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THE LIBRARY

THE UNIVERSITY

OF CALIFORNIA

LOS ANGELES

GIFT

Dr. K. N, Beigelman

' 'm ■■

DPTICKSi

O R, A

TREATISE

OF THE

REFLEXIONS, REFRACTIONS,
INFLEXIONS and COLOURS

O F

L I G U T

ALSO

Two TREATISES

OF THE

SPECIES and MAGNITUDE

O E

I I

Curvilinear Figures

L 0 N D 0 A%
Printed for Sam,. Smith, and Ben J. Walpo^ic©^

Printers to the Royal Society , at tliG PrJMi:''s Arms in
St. FauPs Church-yard. MDCCIV*

-.f^: •■

=-m

p-^ftt-. <■"'*!? '- ^

^ AC
8

ADVERTISEMENT. ^'^],^^

PArt of the enfuing Difcourfe ahout Light was written
at the dejire of J ome Gentlemen of the Royal Society,

in the Tear 1675. ^^^ ^^^^ fi^^ ^^ ^^^^^^ Secretary^ and
read at their Meetings^ and the reft was added about
Twelve Tears after to complete the Theory ; except the
Third Book, and the laft Propo/ition of the Second, which
werejince put together out offcattered Papers. To avoid
heing engaged in Difputes about thefe Matters, I have
hitherto delayed the Printing, and Jhould ftill huve de-
layed it, had not the importunity of Friends prevailed upon
me. If any other Papers writ on this Subject are got out
<f my Hands they are imperfecl,and were perhaps written
before I had tried all the Experiments her^ fet dovjn,
and fully fatisfied my felf about the Laws of Refractions
and Compojition of Colours. I have here Publifhed what
I think proper to come abroad, wifhing that it may not be
Tranftated into another Language without my Confent.

The Crowns of Colours, which fometimes appear about
the Sun and Moon, I have endeavoured to give an Ac-
count of; but for want offufficient Obfer vat ions leave that
Matter to be further examined. The Subject of the Third
Book I have alfo left imperfect, not having trieS all the

Expe-

Kxperhnents zvJucb I intended ivhen I vjas about thefe
Matters, nor repeated fome of thoje ixjhich I did try, until
Ihadjatisfied niyfeJf about all their Circumjlances. To
communicate what I have tried, and leave the reft to
others for further Enquiry, is all my Deji^a in publijhing
thefe Papers.

In a Letter written to Mr.Leibnitz in the Tear 1 6'j6.
and publ/fJoed by Dr. Wallis, I mentioned a Method by
which I had found fome general Theorems about Jquaring
Curvilinear Figures, or comparing them with the Conic
Scilions, or other the Jim pie ft figures with which they may
he compared. And fome Tears ago I lent out a Manufcript
i-ontainingfuch Theorems, and having fince met with fome
Things copied out of it, I have on this Occafton made it
publick, prefixing to it an Introduclion and fubyjyning a
Scholium concerning that Method. And I have pined
with it another fmall Tract concerning the Curvilinear
Figures of the Second Kind, which was alfo written
ntany Tears ago, and made known to fome Friends, who
bavejolktted the making it publick.

I. N.

The FIRST BOOK

O F

O P T I C K S

PART I.

MY D^fign in this Book is not to explain the Pro-
perties of Light by Hypothefes, but to propofc
and prove them by Reafon and Experiments ;
In order to which , I (hall premife the following Defini-
tions and Axioms.

DEFINITIONS,

D E F I N. L

Br the ^ys of Light I nnderftand its leajl Tarts , and thofe
06 wellSucceJJiVe in the fame Lines as Contemporary in Je'
Veral Lines. For it is manifeft that Light confifts of parts
both Succeflive and Contemporary 5 becaufe in the fame
place you may flop that which comes one moment, and
let pafs that which comes prefently afterj and in the fame
time you may ftop it in any one place, and let it pafs in
any other. For that part of Light which is ftopt cannot
be the fame with that which is let pafs. The leaft Light
or part of Light , which may be ftopt alone without the
left of the Light, or propagated alone, or do or fuffer any

A thing

[2]

thing alone, which the reft of the Light doth not or fuf-
ers not, I call a Ray of Light.

D E F I N. 11.

^efrangibil'tty of the ^ys of Lights ts their Difpofition to h
refraHed or turned out of their Way in paffing out of one trmf'
parent 'Body or 'Medium into another. And a greater er lefs (?^e-
frangihility of^ys^ is their Difpofition to he turned more or lefs
out of their Way in like Incidences on the fame Medium. Mathe-
maticians ufually confider the Rays of Light to be Lines
reaching from the luminous Body to the body illumina-
ted, and the refradlion of thofe Rays to be the bending
or breaking of thofe Lines in their paffing out of one Me-
dium into another. And thus may Rays and Refi:ad:ions
be confidered, if Light be propagated in an inftant. But
by an Argument taken from the y£quations of the times
of the Eclipfes of Jupiter'' s Satellites it feems that Light is
propagated in time, fpending in its paflage from the Sun
to us about Seven Minutes ol time : And therefore I have
chofen to define Rays and Refractions in fuch general
terms as may agree to Light in both cafes.

D E F I N. III.

^fexihility of ^ys, is their Difpofition to be turned hac{_ into
the fame Medium from any other Medium upon whofe Surface they
fall. Aid ^ys are more or lef^ reflexible , which are returned
back^ more or lefs eafily. As if Light pafs out of Glafs into
Air , and by being inclined more and more to the com-
mon Surface of the Glafs and Air, begins at length to be
totally reflected by that Surface 5 thofe forts of Rays which
at like Incidences are reflected moft copiouily , or by in-
chning the Rays begin fooneft to be totally reflected, are-
moft reflexible. D E-

[ ? }

D E F I N. IV.

The Angle of Incidence^ is that Angle which the Line defcribed
by the incident ^^y contains with the Perpendicular to the refle"
fiing or refraSiing Surface at the Toint of Incidence.

D E F I N. V.

The Angle of ^flexion or ^fraBion, «■ the Angle which the
Line defcribed by the rejleSled or refraBed ^y containeth with
the perpendicular to the refleBing or refraBing Surface at the
^oint of Incidence.

D E F I N. VI.

The Sines of Incidence, Reflexion, and ^fraBion, are the
Sines of the Angles of Incidence^ Reflexion, and ^efraBion.

D E F I N. VII.

The Light whofe ^ys are all alike ^J^efrangible, I call Sinu
pie , Homogeneal and Similar j and that whofe ^ys are fome
more ^frangible than others , I call Com^otmdj Heteroo-eneal and
'DiJJimilar. The former Light I call Homogeneal , not
becaulc I would affirm it fo in all refpecfls ; but becaufe
the Rays which agree in Refrangibility, agree at leaft ia
all thofe their other Properties. Which I confider in the
following Diicourfe.

D E F I N. VIII.

The Colours of Homogeneal Lights , / call Primary, Homo-
geneal and Simple j and thofe of Heterogeneal Lights, Heteroge'
neal and Compound. For thefe are always compounded of
the colours of Homogeneal Lights 3 as will appear in the
following Difcourfe. A 2 AX I-

C4]
AXIOMS.

A X. I.

THE Angles of Incidence^ ^flexmi, and ^frafiion, lye
ill one and the fatne Tlofie.

A X. II.

The Angle of ^flexion is equal to the Angle of Incidence.

A X. III.

If the refraSied ^y he returned direBly back, fo the Pointy
of Incidence , it JJ^all he refrafied into the Line before defcri-
bed by the incident (I(ay.

A X. IV.

^fraflion out of the rarer Medium into the denfer , is made
towards the Perpendicular 3 that is, fo that the Angle of ^fra-
Bion be leJJ than the Angle of Incidence.

AX. V.

The Sine of Incidence, is either accurately or "Very nearly in a
giyen <B^tio to the Sine of ^fraElion.

Whence if that Proportion be known in any one Incli»
nation of the incident Ray, 'tis known in all the Inclina-
tions, and thereby the Refradion in all cafes of Incidence
on the fame refracting Body may be determined. Thus
if the Refradion be made out of Air into Water, the Sine
of Incidence of the red Light is to the Sine of its Refra-
ction as 4, to ^0. If out of Air into Glafs, the Sines are

C 5 ]

as 17 to 1 1. In Light of other Colours the Sines have
©ther Proportions : but the difference is fo little that if
need feldom be considered.

Suppofe therefore, that R S reprefents the Surface of Fix-
ftagnating Water , and C is the point of Incidence in
which any Ray coming in the Air from A in the Line
A C is reflected or refraded, and I would know whether
this Ray fhall go after Reflexion or Refradlion : I ere(5l
upon the Surface of the Water from the point of Inci-
dence the Perpendicular C P and produce it downwards-
to Q., and conclude by the firfl: Axiom, that the Ray af-
ter Reflexion and Refrad:ion, fhall be found fomewhere in
the Plane of the Angle of Incidence A C P produced. I
let fall therefore upon the Perpendicular C P the Sine of
Incidence A D, and if the reflected Ray be defired , I pro-
duce A D to B fo that D B be equal to A D, and draw
C B. For this Line C B fhall be the refleded Ray; the
Angle of Reflexion B C P and its Sine B D being equal
to the Angle and Sine of Incidence, as they ought to be
by the fecond Axiom. But if the refradled Ray be de-
fired, I produce AD toH, fo that D H may be to A Di
as the Sine of Refra<5tion to the Sine of Incidence, that is
as 3 to 4 ; and about the Center C and in the Plane A C P'
with the Radius C A defcribing a Circle A B E I draw
Parallel to the Perpendicular C P Q, the Line H E cutting'
the circumference in E, and joyning C E, this Line CE
(hall be the Line of the refradied Ray. For if E F be let
fall perpendicularly on the Line P Q_ , this Line E F fhall
be the Sine of Refradion of the Ray C E, the Angle of
Refradion being E C Q; and this Sine E F is equal to
DH, and confequently in Proportion to the Sine oflnci---
dence AD as 3 to 4.

C6]

• In like manner, if there be a Prifm of Glafs (that is a
Glafs bounded with two Equal and Parallel Triangular
ends, and three plane and well polifhed Sides, which meet
in three Parallel Lines running from the three Angles of
one end to the three Angles of the other end) and if the
Refraction of the Light in paffing crofs this Prifm be dtCi'
fi<r. 2. red : Let ACB reprefent a Plane cutting this Prifm tranf-
verfly to its three Parallel lines or edges there where the
Light paffeth through it, and let J E be the R.iy inci-
dent upon the firft fide of the Prifm A C where the Light
goes into the Glafs 3 And by putting the Proportion of
the Sine of Incidence to the Sine of Refracftion as 17 to
1 1 find E F the firft refraded Ray. Then taking this Ray
for the Incident Ray upon the fecond fide of the Glafs B C
where the Light goes out, find the next refradted Ray F G
by putting the Proportion of the Sine of Incidence to the
Sine of Refrad:ion as 1 1 to 17. For if the Sine of Inci-
dence out of Air into Glafs be to the Sine of Refraction
as 17 to 1 1, the Sine of Incidence out of Glafs into Air
muft on the contrary be to the Sine of RefraClion as i i
to 17, by the third Axiom.
fig-. 7 . Much afi:er the fame manner , if A C B D reprefent a
Glafs fpherically Convex on both fides (ufually called a
Lens, fuch as is a Burning- glafs, or Spectaclc-glafs, or an
Objedt-glafs of a Telefcope) and it be required to know
how Light falling upon it from any lucid point Q_ fliall
be refracted, let Q.M reprefent a Ray falling upon any
point M of its firft fpherical Surface ACB, and by ered:ing
a Perpendicular to the Glafs at the point M, find the firft
refracted Ray M N by the Proportion of the Sines 1 7
to 1 1 . Let that Ray in going out of the Glafs be inci-
dent upon N, and then find the fecond refracted Ray N q
by the Proportion of the Sines 1 1 to 1 7. And after the

fame

C7]

fame manner may the Refradlion be found when the
Lens is Convex on one fide and Plane or Concave on
the other, or Concave on both Sides.

AX. VI.

Homogened ^ys which flow from feVeral 'Points of any Oh'
jeSl, and fall almoft ferpendicularly on any refiefiing or refra^
ilrng Tlane or Spherical Surface , Jhall afterwards diverge from
fo many other Joints ^ or he ^Parallel to fo ynany other Lines, or
conserve to fo ynany other 'Points, either accurately or without any
fenfihle Error. And the fame thing will happen, if the ^ys he
refleSied or rcfraSled fucceJJiVely hy two or three or 7?iore 'Playie
or Jpherical Surfaces.

The Point from which Rays diverge or to which they
converge may be called their Focus. And the Focus of
the incident Rays being given, that of the refledted or re-
fraded ones may be found by finding the Refradion of
any two Rays, as above 5 or more readily thus.

Caf 1. Let ACB be a reflecting or refracting Plane, Fig. 4.
and Q. the Focus of the incident Rays, and Q.^ C a per-
pendicular to that Plane. And if this perpendicular be
produced to q, fo that ^ C be equal to Q.C, the point q
fhall be the Focus of the reflected Rays. Or if ^ C be
taken on the fame fide of the Plane with Q_C and in Pro-
portion to Q.C as the Sine of Incidence to the Sine of
Refradiion, the point q fhall be the Focus of the refrac-
ted Rays.

Caf. 2. Let A C B be the reflecting Surface of any Fig. 5.
Sphere whofe Center is E. Bifect any Radius thereof (fup-
pofe E C) in T, and if in that Radius on the fame fide the
point T you take the Points Q. and q, fo that T Q., T E,
and Tq be continual Proportionals, and the point Q.be

the

[8]

the Focus of the incident Rays , the point q (hall be the
Focus of the refled:ecl ones.

Fig. 6. Caf. I . Let A C B be the refracting Surface of any
Sphere whofe Center is E. In any Radius thereof E C
'produced both ways take E T and C t feverally in fuch
Proportion to that Radius as the leflcr of the Sines of
Incidence and Refradlion hath to the difference of thofe
Sines. And then if in the fame Line you find any two
Points Q. and q , fo that T Q. be to E T as E f to f ^,
taking t q the contrary way from t which T Q. Heth from
T, and if the Point Qbe the Focus of any incident Rays,
the Point q fhall be the Focus of the refra(5led ones.

And by the fame means the Focus of the Rays after
two or more Reflexions or Refractions may be found.

^jFtg. 7 . Caf. 4. Let A C B D be any refradling Lens , fpheri-
cally Convex or Concave or Plane on either fide , and let
C D be its Axis (that is the Line which cuts both its Sur-
faces perpendicularly, and pafles through the Centers of
the Spheres,) and in this Axis let F and /be the Foci of the
refracted Rays found as above , when the incident Rays
on both fides the Lens are Parallel to the fame Axis 3 and
upon the Diameter F/ bifected in E, defcribe a Circle.
Suppofe novj that any Point Q. be the Focus of any inci-
dent Ray So Draw Q,E cutting the faid Circle in T and f,
and therein take t q inilich Proportion to f E as f E or TE
hath to T Q. Let t q lye the contrary way from t which
T Q. doth from T, and q fhall be the Focus of the refrac-
ted Rays without any fenfible Error , provided the Point
Q_ be not fo remote from the Axis, nor the Lens fo broad
as to make any of the Rays fall too obliquely on the
refracting Surfaces.

And by the like Operations may the reflecting or re-
fracting Surfaces be found when the two Foci are given,

and

[ 9 ]

and thereby a Lens be formed, which (liall make the Rayt
flow towards or from what place you pleafe.

So then the meaning of this Axiom is , that if Ray$
fall upon any Plane or Spherical Surface or Lens, and
before their Incidence flow from or towards any Point Q. ,
they (hall after Reflexion or Refraction flow from or to-
wards the Point q found by the foregoing Rules. And if
the incident Rays flow from or towards feveral points Q.,
the reflected or refracted Rays fball flow from or towards
To many other Points q found by the fame Rules. Whe-
ther the reflected and refracted Rays flow from or towards
the Point q is eafily known by the fituation of that Point.
For if that Point be on the fame fide of the reflecting or
refracting Surface or Lens with the Point Q, and the in-
cident Rays flow from the Point Q, the refle(5ted flow to-
wards the Point q and the refracted from it 5 and if the
incident Rays flow towards Q, the reflected flow from ^,
and the refracted towards it. And the contrary happens
when q is on the other fide of that Surface.

A X. VII.

Wherever the ^ys which come from all the Joints of any Ob'
jeFi yneet again in fo many joints after they haVe been ?nade to
converge by Reflexion or ^efraflion^ there they tvill make a Tic'
ture of the Object upon any white 'Body on which they fall.

So if PR reprefent any Object without Doors, and ABFig.
be a Lens placed at a hole in the Window-fhut of a dark
Chamber, whereby the Rays that come from any Point Q_
of that Object are made to converge and meet again in
the Point q 5 and if a Sheet of white Paper be held at q
for the Light there to fall upon it : the Picture of that
Object PR will appear upon the Paper in its proper Shape

B and

[lo]

and Colours. For as the Light which comes from the
Point Q_ goes to the Point q, [o the Light which comes
from orhcr Points P and R of the Object, will go to fo
many other correfpondent Points fi and r (as is manifeft
by the fixth Axiom 3) fo chat every Point of the Objecr
fhall illuminate a correfpondent Point of the Picture, and
thereby make a Picture like the Object in Shape and Co-
lour, this only excepted that the Picture fhall be inverted.
And this is the reafon of that Vule,ar Experiment of call-
ing the Species of Objects from abroad upon a Wall 01
Sheet of white Paper in a dark Room.
8. In like manner when a Man viev.s any Object P Q.R,
the Light which comes from the fevcral Points ol the Ob-
ject is fo refracted by the tranlparenc skins and humours
of the Eye, (that is by the outward coat EFG called the
Tunica Comen, and by the cryftalline humour AB which is
beyond the Pupil m k^) as to converge and meet again at
fo many Points in the bottom of the Eye, and there to paint
the Picture of the Object upon that skin (called the T«-
nica ^tina) with which the bottom of the Eye is covered.
For Anatomifts when they have taken off from the bot-
tom of the Eye that outward and moft thick Coat called
the Dura Matey, can then fee through the thinner Coats
the Pictures of Objects lively painted thereon. And thefe
Pictures propagated by Motion along the Fibres of the Op-
tick Nerves into the Brain, are the caufe of Vifion. For
accordingly as thefe Pictures are perfect or imperfect, the
Object is feen perfectly or imperfectly. If the Eye be tin-
ged with any colour Cas in the Difeafe of the Jaundi/e) fo
as to tinge the Pictures m the bottom of the Eye with that
Colour, then all Objects appear tinged with the fame Co-
lour. If the humours of the Eye by old Age decay, fo
45 by llirinking to make the Coni£a and Coat of the Cry'

, Jlalline

["]

flalUne humour grow flatter than before, the Light will not be
refracted enough , and for want of a fufficient Refradion
will not converge to the bottom of the Eye but to fome
place beyond it , and by confequence paint in the bottom
of the Eye aconfufedPi(5ture,and according to the indiftindt-
nefs of this Piifture the Objedt will appear confufed. This
is the reafon of the decay of Sight in old Men, and fliews
why their Sight is mended by Spedacles. Forthofe Con-
vex-glafles fupply the defedl of plumpnefs in the Eye, and
by encreafing the Refraction make theRays converge fooner
fo as to convene diftincftly at the bottom of the Eye if the
Glafs have a due degree of convexity. And the contrary
happens in fhort-fighted Men whofe Eyes are too plump.
For the Refra<5tion being now too great, the Rays converge
and convene in the Eyes before they come at the bottom 5
and therefore thePiClure made in the bottom and the Vifion
caufed thereby will not be diftind, unlefs the Objed be
brought fo near the Eye as that the place where the con-
verging Rays convene may be removed to the bottom, or
that the plumpnefs of the Eye be taken off and the Refra-
diions diminifhed by a Concave-glafs of a due degree of
Concavity, or laftly that by Age the Eye grow flatter till it
come to a due Figure : For fhort-fighccd Men fee remote
Objeds beft in Old Age, and therefore they are accounted
to have the moft laftmg Eyes.

A X. VIII.

An Ohjefi feen by Reflexion or ^fraSlion^ appears in that place
from iphence the ^ys after their lajl ^flexion or l^frafiion di'
yer^e in falling on the SpeHators Eye.

If the Objed A be feen by Reflexion of a Looking- Fig. ^\
glafs m ?i, it fliall appear, not in it's proper place A, but

B 2 behind

I-

[12]

behind the Glafs at 4, from whence any Rays AB, AC,
A D, which flow from one and the fame Point of the Ob*
jed, do after their Reflexion made in the Points B, C, D,
diverge in going from the Glafs to E, F, G, where they
are incident on the Spectator's Eyes, For thefe Rays do
make the fame Picture in the bottom of the Eyes as if
they had come from the Object really placed at a, without
the interpofition of the Looking-glafs j and all Vifion is
made according to the place and fhape of that Picture.

F/C-, 2. In like manner the Object D feen through a Prifm ap-
pears not in its proper place D, but is thence tranflated to
ibme other place d fituated in the lafl refracted Ray F G
drawn backward from F to d.

Til. I o. And fo the Object Q. feen through the Lens A B, appean
at the place q from whence the Rays diverge in pafling
from the Lens to the Eye. Now it is to be noted, that the
Image of the Object at q is fo much bigger or leffer than
the Object it felf at Q., as the diftance of the Image at
q from the Lens AB is bigger or lefs than the diftance of
the Object at Q. from the fame Lens. And if the Object
be feen through two or more fuch Convex or Concave-
glaffes, every Glafs (hall make a new Image, and the Ob-
jed fhall appear in the place and of the bignefs of the laft
Image. Which confideration unfolds the Theory of Mi-
erofcopes and Telefcopes. For that Theory confiils in al-
moft nothing elfe than the defcribing fuch Glafles as fliall
make the laft Image of any Obje(5t as diftindl and large
and luminous as it can conveniently be made.

I have now given in Axioms and their Explications the
fumm of what hath hitherto been treated of in Opticks.
For what hath been generally agreed on I content my
felf to afliime under the notion of Principles, in order to
what I have further to write. And this may fuffice for an:

Intro-

[13]

Introdudion to Readers of quick Wit and good Under-
ftanding not yet verfed in Opticks : Although thofe who
are already acquainted with this Science , and have
handled Glafles, will more readily apprehend what fol^
loweth.

PROPOSITIONS.

(p^I^OT. I. Theor. I.

I G H T S which differ in Colour, differ alfo in De-
grees of Refrangibility.

The Proof hy E>cperiments.

Exper. I . I took a black oblong ftiff Paper terminated
by Parallel Sides, and with a Perpendicular right Line
drawn crofs from one Side to the other , diftinguifhed it
into two equal Parts. One of thefe Parts 1 painted with
a red Colour and the other with a blew. The Paper was
very black, and the Colours intenfe and thickly laid on,
that the Phacnomenon might be more confpicuous. This
Paper I viewed through a Prifm of folid Glafs, whofe two
Sides through which the Light paffed to the Eye were
plane and well polifhed, and contained an Angle of about
Sixty Degrees : which Angle I call the refrad:ing Angle of
the Prifm. And whilft I viewed it, I held it before a
Window in fuch manner that the Sides of the Paper were
parallel to the Prifm, and both thofe Sides and the Prifm
parallel to the Horizon, and the crofs Line perpendicular
to it ; and. that the Light which fell from the Window

UpOft

117. I 1

[Hi

upon Vhe Piper made an Angle with the Paper, equal to
that Angle which was made with the fame Paper by the
.Light refieiled from it to the Eye. Beyond the Prifm was
the Wall of the Chamber under the Window covered over
with black Cloth, and the Cloth was involved in Dark-
nefs that no Light might be refleded from thence, which
in pafling by the edges of the Paper to the Eye , might
mingle it felf with the Light of the Paper and obfcure the
Phacnomenon thereof Thefe things being thus ordered,
I found that if the refrading Angle of the Prifm be turned
upwards, fo that the Paper may feem to be lifted upwards
by the Refradion , its blew half will be lifted higher by
the Refradion than its red half But if the refracting
Angle of the Prifm be turned downward, fo that the Pa-
per may feem to be carried lower by the Refradlion, its
blew half will be carried fomething lower thereby than
its red half Wherefore in both cafes the Light v/hich
comes from the blew half of the Paper through the Prifm
to the Eye, does in like Circumftances fuffcr a greater Re-
fra(5lion than the Light which comes from the red half,
and by confequence is more refrangible.

lUuftratiGn. In the Eleventh Figure, M "N reprefents the
Window,and D E the Paper terminated with parallel Sides
D J and H E, and by the tranfverfe Line F G diftinguilTied
into two halfs, the one D G of an intenfely blew Colour,
the other F Eof an intenfely red. And '^KCcab repre-
fents the Prifm whofe refrading Planes h^b a and hQca
meet in the edge of the refrading Angle A a. This edge
A^ being upward, is parallel both to the Horizon and to
the parallel edges of the Paper D J and H E. And de re-
prefents the Image of the Paper feen by Refraction up-
wards in fuch manner that the blew half D G is carried
higher to d^ than the red half F E is to /e, and therefore"

fuffers

[15]

fufFers a greater Rcfradion. If the edge of the refracting
Angle be turned downward, the Image of the Paper will
be refraded downward fuppofe to ^i, and the blew half
will be refraded lower to -^ 7 than the red half is to ?>s.

Exper. 2. About the afotefaid Paper, whofe two halfs
were painted' over with red and blew, and which was ftiff
like thin Pailboard, I lapped feierai times a llender thred
of very black Silk, in fuch manner that the feveral parts
of the thred might appear upon the Colours like fo many
black Lines drawn over them , or like long and llender
dark Shadows caft upon them. I might have drawn black
Lines with a Pen, but the threds were fmaller and better
defined. This Paper thus coloured and lined i let againft
a Wall perpendicularly to the Horizon, fo that one of the
Colours might ftand to the right hand and the other to
the left. Clofe before the Paper at the confine of the Co-
lours below I placed a Candle to illuminate the Paper
ftrongly : For the Experiment was tried in the Night.
The flame of the Candle reached up to the lower edge of
the Paper, or a very little higher. Then at the diftance of
Six Feet and one or two Inches from the Paper upon the
Floor I eredled a glafs Lens four Inches and a quarter
broad, which might colled the Rays coming from the
feveral Points of the Paper, and make them converge to-
wards fo many other Points at the fame diftance of fix
Feet and one or two Inches on the other fide of the Lens,
and fo form the Image of the coloured Paper upon a white
Paper placed there 3 after the fame manner that a Lens at
a hole in a Window cafts the Images of Objeds abroad
upon a Sheet of white Paper in a dark Room. The afore-
faid white Paper, erecled perpendicular to the Horizon
and to the Rays Vv'hich fell upon it firom the Lens, I moved
fometimes towards the Lens, fometimes from it, to find

the

[i6]

•tlie places where the Images of the blew and red parts of
the coloured Paper appeared moft diftind. Thofe places
1 eafily knew by the Images of the black Lines which I
had made by winding the Silk about the Paper. For the
Images of thofe fine and flender Lines (which by reafon of
their blacknefs were like Shadows on the Colours) were
confijfed and fcarce vifible, unlefs when the Colours on ei-
ther fide of each Line were terminated moft diftindily.
Noting therefore, as diligently as I could, the places where
the Images of the red and blew halfs of the coloured Pa-
per appeared moft diftincft , I found that where the red
half of the Paper appeared diftin(5t, the blew half appeared
confufed, fo that the black Lines drawn upon it could
fcarce be feen 5 and on the contrary where the blew half
appeared moft diftinct the red half appeared confufed, fo
that the black Lines upon it were fcarce vifible. And be-
tween the two places where thefc Images appeared diftind:
there was the diftance of an Inch and a hail : the diftance
of the white Paper from the Lens, when the Image of the
red half of the coloured Paper appeared moft diftind:, be-
ing greater by an Inch and an half than the diftance of the
fame white Paper from the Lens when the Image of the
blew half appeared moft diftintft. In like Incidences there-
fore of the blew and red upon the Lens, the blew was re-
fradted more by the Lens than the red, fo as to converge
fooner by an Inch and an half, and therefore is more refran-
gible.
Pi?. 12. Illuftration. In the Twelfth Figure, DE fignifies the co-
loured Paper, D G the blew half, F E the red half, M N
the Lens, H J the white Paper in that place where the red
half wirh its black Lines appeared diftintfl, and hi the fame
Paper in that place where the blew half appeared diftind:.
The place hi was nearer to the Lens M N than the place
H J by an Inch and an half. Scholtum.

[17]

Scholium. The fame things fucceed notvvithflanding that
fome of the Circumftances be varied : as in the firft Ex-
periment when the Pnfm and Paper are any uays inclined
to the Horizon , and in both when coloured Lines are
drawn upon very black Paper. But in the Defcription of
thefe Experiments , I have fet down fuch Circumftances
by which either the Phaenomenon might be rendred more
confpicuous, or a Novice might more eafily try them, or
by which I did try them only. The fame thing I have
often done in the following Experiments : Concerning all
which this one Admonition may fuffice. Now from thefe
Experiments it follows not that all the Light of the blew
is more Refrangible than all the Light of the red 3 For
both Lights are mixed of Rays differently Refrangible,
So that in the red there are fome Rays not lefs Refrangible
than thofe of the blew , and in the blew there are iome
Rays not more Refrangible than thofe of the red j But
thefe Rays in Proportion to the whole Light are but feWj
and ferve to diminifli the Event of the Experiment , but
are not able to deftroy it. For if the red and blew Co-
lours were more dilute and weak, the diftance of the Ima-
ges would be lefs than an Inch and an half 5 and if they
were more intenfe and full, that diftance would be greater,
as will appear hereafter. Thefe Experiments may fuffice
for the Colours of Natural Bodies. For in the Colours
made by the Refraction of Prifms this Propofition will
appear by the Experiments which are now to follow in the
next Propofition.

TfJlOT.

[i8]
PROP. II. Theor. II.

The Light of the Sun confifis of ^ys differently ^frangible.

The Proof by Experiments.

Exper. 3. TN a very dark Chamber at a round hole about
j[ one third part of an Inch broad made in the
Shut of a Window I placed a Glafs Prifm, whereby the
beam of the Sun's Light which came in at that hole might
be refrad:ed upwards toward the oppofite Wall of the
Chamber , and there form a coloured Image of the
Sun. The Axis of the Prifm (that is the Line paifing
through the middle of the Prifm from one end of it to
the other end Parallel to the edge of the Refrading Angle)
was in this and the following Experiments perpendicular
to the incident Rays. About this Axis I turned the Prifm
flowly , and faw the refra(5l:ed Light on the Wall or co-
loured Image of the Sun firft to defcend and then to af-
cend. Between the Defcent and Afcent when the Image
feemed Stationary , I ftopt the Prifm, and fixt it in that
Pofture, that it fliould be moved no more. For in that
poflure the Refractions of the Light at the two fides of
the Refra<5ting Angle, that is at the entrance of the Rays
into the Prifm and at their going out of it, were equal to
one another. So alfo in other Experiments as often as I
would have the Refractions on both fides the Prifm to be
equal to one another, I noted the place where the Image
of the Sun formed by the refrad:ed Light flood ftill be-
tween its two contrary Motions, in the common Period
of its progrefs and egrefs 5 and when the Image fell upon
that plaeCj I made faft the Prifin. And in this pofture, as

the

[19]

the mod convenient,ic is to be underftood that all the Prifms
are placed in the following Experiments, unlefs where feme
other pofture is defcribed. The Prifm therefore being pla-
ced in this pofture, I let the refra<5ted Light fall perpendi-
cularly upon a Sheet of white Paper at the oppofite Wall
of the Chamber, and obferved the Figure and Dimenfions
of the Solar Image formed on the Paper by that Light.
This Image was Oblong and not Oval, but terminated
with two Rectilinear and Parallel Sides , and two Semi-
circular Ends. On its Sides it was bounded pretty diftindly,
but on its Ends very confufedly and indiftindlly, the Light
there decaying and vanifhing by degrees. The breadth of
this Image anfwered to the Sun's Diameter, and was about
two Inches and the eighth part of an Inch , including the
Penumbra. For the Image was eighteen Feet and an half
diftant from the Prifm, and at this diftance that breadth if
diminiflied by the Diameter of the hole in the Window-fliut,
that is by a quarter of an Inch, fubtended an Angle at the
Prifm of about half a Degree, which is the Sun's apparent
Diameter. But the length of the Image was about ten Inches
and a quarter, and the length of the Redilinear Sides about
eight Inches 5 And the refracting Angle of the Prifm where-
by fo great a length v/as made, was 64 degr. With a lefs
Angle the length of the Image was lefs , the breadth re-
maining the fame* If the Prifm was turned about its Axis
that way which made the Rays emerge more obliquely out
of the fecond refraCtino; Surface of the Prifm, the Imase foon
became an Inch or two longer, or more; and if the Prifm
was turned about the contrary way, fo as to mal^e the Rays
fall more obliquely on the firft refracting Surface, the Image
foon became an Inch or two fhorter. And therefore in try-
ing this Experiment, I was as curious as I could be in pla-
cing the Prifm by the above-mentioned Rule exaCtly in

C 2 fuch

[20]

fuch a pofture that the Refradions of the Rays at their emer-'
gence out of the Prifm might be equal to that at their inci"
dence on it. This Prifm had fome Veins running along
within the Glafs from one end to the other , which feat-
tercd fome of the Sun's Light irregularly, but had no fen-
hble effed; in encreafing the length of the coloured Spec-
trum. For I tried the fame Experiment with other Prifms
with the fame Succefs. And particularly with a Prifm
which feemed free from fuch Veins, and whofe refracting
Angle was 6i\ Degrees, I found the length of the Image 9^

or 10 Inches at the diftance of 18- Feet from the Prifm,
the breadth of the hole in the Window-fhut being i of an

Inch as before. And becaufe it is eafie to commit a mi-
flake in placing the Prifm in its due pofture, I repeated
the Experiment four or five times, and always found the
length of the Image that which is fet down above. With
another Prifm of clearer Glafs and better PoUifh, which
feemed free from Veins and whofe refracting Angle was
63 ' Degrees, the length of this Image at the fame diftance
of I 8 ^ Feet was alfo about 1 o Inches, or 10^. Beyond

thefe Meafures for about ' or - of an Inch at either end of

4 3

the Spe6trum the Light of the Clouds feemed to be a little
tinged with red and violet, but fo very faintly that I fufpe-
d:ed that tinCture might either wholly or in great meafure
arife from fome Rays of the SpeCtrum fcattered irre-
gularly by fome inequalities in the Subftance and Polifh
of the Glafs , and therefore I did not include it in thefe
Meafures. Now the different Magnitude of the hole in
theWindow-fliut, and different thicknefs of the Prifm where
the Rays paffed through it, and different inclinations of the
Prifm to the Horizon, made no fenfible changes in the
kngtK of the Image. Neither did the different matter of ^

the

[21]

the Prifms make any : for in a Veflel made of poliflied-
Plates of Glafs cemented together in the (hape of a Prifm
and filled with Water, there is the like Succefs of the Ex-
periment according to the quantity of the Refradiion. It
is fijrther to be obferved, that the Rays went on in right
Lines from the Prifm to the Image, and therefore at their
very going out of the Prifm had all that Inclination to
one another from which the length of the Image pro-
ceeded, that is the Inclination of more than two Degrees
and an half And yet according to the Laws of Opticks
vulgarly received, they could not poifibly be fo much in-
clined to one another. For let EG reprefent the Window- fm-. i ?,
(hut, F the hole made therein through which a beam of the
Sun's Light was tranfmitted into the darkned Chamber, and
ABC a Triangular Imaginary Plane whereby the Prifm is
feigned to be cut tranfverfly through the middle of the
Light. Or if you pleafe, let A B C reprefent the Prifm it
felf, looking diretflly towards the Spectator's Eye with its
nearer end : And let X Y be the Sun, MN the Paper upon
which the Solar Image or Spectrum is caft, and P T the
Image it felf whofe fides towards V and W are ReClili-
near and Parallel, and ends towards P and T Semicir*
cular. Y K H P and X L J T are the two Rays, the firft
of which comes from the lower part of the Sun to the
higher part of the Image, and is refracted in the Prifm at
K and H, and the latter comes from the higher part of
the Sun to the lower part of the Image, and is refraded
at L and J. Since the Refrad:ions on both fides the Prifm
are equal to one another, that is the Refradion at K equal
to the Refradion at J, and the Refradion at L equal ta
the Refradion at H, fo that the Refradions of the inci-
dent Rays at K and L taken together are equal to the
Refradions of the emergent Rays at H and J taken toge-
ther ;.

[22]

ther : it follows by adding equal things to equal things,
that the Refradions at K and H taken together, are equal
to the Refrad:ions at J and L taken together , and there-
fore the two Rays being equally refracted have the fame
Inclination to one another after Refrad;ion which they had
before, that is the Inclination of half a Degree anfwering
to the Sun's Diameter. For fo great was the Inclination
of the Rays to one another before Refradion. So then,
the length of the Image P T would by the Rules of Vul-
gar Opticks fubtend an Angle of half a Degree at the
Prifm, and by confequence be equal to the breadth > v> ;
and therefore the Image would be round. Thus it would
be were the two Rays X L J T and Y K H P and all the
reft which form the Image P jp T >, alike Refrangible.
And therefore feeing by Experience it is found that the
image is not round but about five times longer than
broad, the Rays which going to the upper end P of the
Image fuffer the greateft Refraction, muft be more Refran-
gible than thofe which go to the lower end T , unlefs the
inequality of Refrad:ion be cafual.

This Image or Sped:rum P T was coloured, being red
at its leaft refracfted end T, and violet at its moft refi:a<5ted
end P, and yellow green and blew in the intermediate
ipaces. Which agrees with the firft Propofition, that Lights
which differ in Colour do alfo differ in Refrangibiiity.
The length of the Image in the foregoing Experiments I
meafured from the faintefl and outmoft red at one end, to
the faintefl and outmofl blew at the other end.

Exper. 4. In the Sun's beam which was propagated in-
to the Room through the hole in the Window-fhut, at
the diftance of fome Feet from the hole, I held the Prifm
in fuch a poflure that its Axis might be perpendicular to
that beam. Then I looked through the Prifm upon the ,

hole,

[23] ■

hole, and turning the Prifm to and fro about its Axis to
make the Image of the hole afcend and defcend, when be-
tween its two contrary Motions it feemed ftationary, I
ftopt the Prifm that the Refradlions on both fides of the
refradling Angle might be equal to each other as in the
former Experiment. In this Situation of the Prifm view-
ing through it the faid hole, I obferved the length of its
refracted Image to be many times greater than its breadth,
and that the moft refracted part thereof appeared violet,
the leaft refradted red, the middle parts blew green and
yellow in order. The fame thing happened when I re-
moved the Prifm out of the Sun s Light , and looked
through it upon the hole fhining by the Light of the
Clouds beyond it. And yet if the Refradlion were done
regularly according to one certain Proportion of the Sines
of Incidence and Refracftion as is vulgarly fuppofed, the
refracted Image ought to have appeared rouna.

So then, by thefe two Experiments it appears that in
equal Incidences there is a confiderable inequality of Re*
fradiions : But whence this inequality arifes, whether it be
that fome of the incident Rays are refrad:ed more and
others lefs, conftantly or by chance, or that one and the
fame Ray is by Refra6tion difturbed, fhattered, dilated,
and as it were fplit and fpread into many diverging Rays,
as GrimaUo fuppofes, does not yet appear by thefe Experi-
ments, but will appear by thofe that follow.

Exper. 5 . Confidering therefore, that if in the third Ex-
periment the Image of the Sun fhould be drawn out into
an oblong form, either by a Dilatation of every Ray, or
by any other cafual inequality of the Refradions, the fame
oblong Image would by a fecond Refradion made Side-
ways be drawn out as much in breadth by the like Dila-
tation of the Rays or other cafual inequality of the Rc-

frad:ions

[24]

iradions Sideways, I tried what would be the EfFctfls of

fiich a fecond Refra6lion. For this end I ordered all thinas

as in the third Experiment, and then placed a fecond Prifm

immediately after the firft in a crofs Pofition to it, that it

might again refra(5l the beam of the Sun s Light which

came to it through the firfl: Prifm. In the firft Prifm this

beam was refracted upwards, and in the fecond Sideways.

And I found that by the Refradlion of the fecond Prifm

the breadth of the Image was not increafed, but its fupe-

rior part which in the firft Prifm fuftered the greater Re-

fi-aition and appeared violet and blew, did again in the

fecond Prifm fuffer a greater Refradlion than its inferior

part, which appeared red and yellow , and this without

any Dilation of the Image in breadth.

fig. 1 4. lUuJlration. Let S reprefent the Sun, F the hole in the

Window, A B C the firft Prifm, D H the fecond Prifm, Y

the round Image of the Sun made by a direcft beam of

Lit^ht when the Prifms are taken away, P T the oblong

Image of the Sun made by that beam paffing through the

£ift Prifm alone when the fecond Prifm is taken away, and

pt the Image made by the crofs Refractions of both

Prifms together. Now if the Rays which tend towards

the feveral Points of the round Image Y were dilated and

fpread by the Refradion of the firft Prifm, fo that they

ITiould not any longer go in fingle Lines to fingle Points,

but that every Ray being fplit, fhattered, and changed

from a Linear Ray to a Superficies of Rays diverging

from the Point of Refraction, and lying in the Plane of

the Angles of Incidence and Refraction, they fliould

go in thofe Planes to fo many Lines reaching almoft

from one end of the Image P T to the other, and if

that Image fhould thence become oblong : thofe Rays

and their feveral parts tending towards the feveral Points qf

the

[25]
the Image P T ought to be again dilated and fpread Side-
ways by the tranfverfe Refraction of the fecond Prifm , fo
as to compofe a fourfquare Image, fuch as is reprefented
at t7. For the better underftanding of which, let the hiiage
FT be diftinguifhed into five equal Parts PQ.K, KQ^RL,
L R S M, M S V N, N V T. And by the fame irregularity
that the Orbicular Light Y is by the Refradion of the firft
Prifm dilated and drawn out into a long Image P T, the
the Light P Q.K which takes up a fpace of the fame length
and breadth with the Light Y ought to be by the Refra-
(Stion of the fecond Prifm dilated and drawn out into the
long Image -rq 4^ ^rid the Light K Q_R L into the long
Image kqrl, and the Lights LRSM, MSVN,NVT
into fo many other long Images I r s 7n, m s y n, nv tl -^ and
all thefe long Images would compofe the fourfquare Image
■^1. Thus it ought to be were every Ray dilated by Re-
fra<flion, and fpread into a triangular Superficies of Rays
diverging from the Point of Refra6tion. For the fecond
Refradiion would fpread the Rays one way as much as the
firfl doth another , and fo dilate the Image in breadth as
much as the firft doth in length. And the fame thing
ought to happen, were fome Rays cafually refra6led more
than others. But the Event is otherwife. The Image P T
was not made broader by the Refraction of the fecond
Prifm, but only became obHque, as 'tis reprefented ztpt,
its upper end P being by the RefraAion tranflated to a
greater diftance than its lower end T. So then the Light
which went towards the upper end P of the Image, was
(at equal Incidences) more refraded in the fecond Prifm
than the Light which tended towards the lower end T,
that is the blew and violet, than the red and yellow j and
therefore was more Refrangible. The fame Light was by
the Refradion of the firft Prifm tranflated further from the

D place

[26]

place Y to which it tended before RcFradion 3 and there-
' fore fuffered as well in the firft Prifm as in the fecond a
greater Refraction than the reft of the Light, and by con-
lequence was more Refrangible than the reft, even before
its incidence on the firft Prifm.

Sometimes I placed a third Prifm after the fecond, and
fometimes alfo a fourth after the third , by all which the
Image might be often refraded fideways : but the Rays
which were more refradcd than the reft in the firft Prifm
were alfo more refradted in all the reft, and that without
any Dilatation of the Image fideways : and therefore thofe
Rays for their conftancy of a greater Refra(5tion are de-
fervedly reputed more Refrangible.
Mig. 15. But that the meaning of this Experiment may more
clearly appear, it is to be confidered that the Rays which
are equally Refrangible do fall upon a circle anfwering to
the Sun's Difque. For this was proved in the third Experi-
ment. By a circle I underftand not here a perfect Geo-
metrical Circle, but any Orbicular Figure whofe length is
equal to its breadth, and which, as to fenfe, may feem
circular. Let therefore A G reprefent the circle which all
the moft Refrangible Rays propagated from the whole
Difque of the Sun, would illuminate and paint upon the
oppofite Wall if they were alone ; E L the circle which all
the leaft Refrangible Rays would in like manner illuminate
and paint if they were alone ; B H, C J, D K, the circles
which fo many intermediate forts of Rays would fuccef-
fively paint upon the Wall, if they were fingly propagated
from the Sun in fucceflive Order, the reft being always in-
tercepted J And conceive that there are other intermediate
Circles without number which innumerable other inter-
mediate forts of Rays would fucceflively paint upon the
Wall if the Sun fliould fucceflively emit every fort apart.

And

[27]

And feeing the Sun emits all thefe forts at once, they muft
ail together illuminate and paint innumerable equal cir-
cles, of all which, being according to their degrees of Re-
frangibility placed in order in a continual fcries, that ob-
long Spedtrum P T is compofed which I defcribed in the
third Experiment. Now if the Sun's circular Image Y
which is made by an unrefrad:ed beam of Light was by
any dilatation of the fingle Rays, or by any other irregu-
larity in the Refraction of the firll Prifm, converted into
the Oblong Spectrum, P T : then ought every circle A G,
B H, C J, <^c. in that Spectrum, by the crofs Refra-
ction of the fecond Prifm again dilating or otherwife
fcattering the Rays as before, to be in like manner drawn
out and transformed into an Oblong Figure, and thereby
the breadth of the Image P T would be now as much aug-
mented as the length of the Image Y was before by the Re-
fraction of the firft Prifm 3 and thus by the Refradions of
both Prifms together would be formed a fourfquare Figure
p"^ tl as I defcribed above. Wherefore fince the breadth of
the Spedtrum P T is not increafed by the Refraction fide-
ways, it is certain that the Rays are not fplit or dilated, or
otherways irregularly fcattered by that Refracftion, but
that every circle is by a regular and uniform Refraction
tranOated entire into another place, as the circle A G by
the greateft RefraCtion into the place ag^ the circle B H by
a lefs Refraction into the place bh^ the circle C J by a Re-
fraction ftill lefs into the place c/, and fo of the refl^ by
which means a new SpeCtrum p t inclined to the former
P T is in like manner compofed of circles lying in a
right Line 5 and thefe circles muft be of the fame bignefs
with the former, becaufe the breadths of all the Spe-
Ctrums. Y, P T and pt at equal diflances from the Prifms'
arc equal.

D 2 I con-

[28]

I confidered further that by the breadth of the hole F
through which the Light enters into the Dark Chamber,
there is a Penumbra made in the circuit of the Spedlrum
Y, and that Penumbra remains in the rectilinear Sides of
the Spedirums P T and pt. I placed therefore at that hole
a Lens or Objecl-glafs of a Telefcope which might caft
the Image of the Sun diftindly on Y without any Penum-
bra at all, and found that the Penumbra of the Rectili-
near Sides of the oblong Spedrums P T and pt was alfo
thereby taken away, fo that thofe Sides appeared as di-
ftinCtly defined as did the Circumference of the firft Image
Y. Thus it happens if the Glafs of the Prifms be free
from veins, atd their Sides be accurately plane and well
polidied without thofe numberlefs waves or curies which,
ufually arife from Sand-holes a little fmoothed in polifli-
ing with Putty. If the Glafs be only well polifbed and.
free from veins and the Sides not accurately plane but a
little Convex or Concave, as it frequently happens 5 yet
may the three Spe6trums Y, P T and pt want Penumbras,
but not in equal diflances from the Prifms. Now from
this want of Penumbras, I knew more certainly that every
one of the circles was refra(5ted according to fome moft
regular, uniform, and conftant law. For if there were
any irregularity in theRefradtion, the right Lines A E and
G L which all the circles in the Spe(5trum P T do touch,
could not by that Refraction be tranflated into the Lines
a e and g I as diftinCt and ftraight as they were before, but
there would arife in thofe tranflated Lines fome Penumbra
or crookednefs or undulation, or other fenfible Perturba-
tion contrary to what is found by Experience. Whatfo-
cver Penumbra or Perturbation fliould be made in the
circles by the crofs Refradion of the fecond Prifm , all
that Penumbra or Perturbation would be confpicuous in

the

[29]
the right Lines a e and g I which touch thofc circles. And
therefore fince there is no fuch Penumbra or Perturbation
in thofe right Lines there mufl be none in the circles.
Since the diftance between thofc Tangents or breadth of
the Spedrum is not increafed by the Refrad:ions, the Dia-
meters of the circles are not increafed thereby. Since thofc
Tangents continue to be right Lines , every circle which
in the firfl; Prifm is more or lefs refrad:ed , is exadlly in-
the fame Proportion more or lefs refra(5led in the fecond.
And leeing all thefe things continue to fucceed after the
fame manner when the Rays are again in a third Prifm,'
and again in a fourth refracted Sideways, it is evident that
the Rays of one and the fame circle as to their degree ob
Refrangibility continue always Uniform and Homogeneal-
to one another, and that thofe of feveral circles do differ
in degree of Refrangibility, and that in fome certain and-
conftant Proportion. Which is the thing I was to prove.

There is yet another Circumftance or two of this Ex-F/^. i6,
perimem by which it becomes flill more plain and con-
vincing. Let the fecond Prifm D H be placed not imme-
ately after after the firft, but at fome diftance from it 5
Suppofe in the mid-way between it and the Wall on which
the oblong Spedlrum P T is caft, fo that the Light from
thefirfl: Prifm may fall upon it in the form of an oblong
Spcdrum, t7 Parallel to this fecond Prifm,and be refraded -
Sideways to form the oblong Spedrum j? t upon the Wall.
And you will find as before, that this Spedrum ^ f is in-
clined to that Spedtrum P T, which the hrft Prifm forms-
alone without the fecond ; the blew ends P and p beina fur-
ther diftant from one another than the red ones T and t,
and by confequence that the Rays which ctq to the blew
end '^ of the Image ■^l and which therefore fuffer the greateftr
Refradion in the firft Prifm, are again in the fecond Prifm
more refraded than the reft. The

C 30 ]

F/g-. 1 7. The fame thing I try'd alTo by letting the Sun's Light
into a dark Room through two little round holes F and p
made in the Window, and with two Parallel Prifms ABC
and A^y placed at thofe holes ( one at each ) refrad:ing
thofe two beams of Light to the oppofite Wall of the
Chamber, in fuch manner chat the two colour'd Images
P T and m n which they there painted were joyned end to
end and lay in one ftraight Line, the red end T of the
one couching the blew end m of the other. For if thefe
two refrad:ed beams were again by a third Prifm D H pla-
ced croft to the two firft, refratied Sideways, and the Spe-
<5trums thereby tranflated to lomc other part of the Wall
of the Chamber , fuppofe the Sp-^flrum P T to pt and
the Spe(5trum M N to m ?i, thefe iranflated Spe(5trums /> t
and m n would not lie in one ftraight Line with their ends
contiguous as before, but be broken off from one another
and become Parallel, the blew end of the Image m n being
by a greater Refradion tranflated farther from its former
place M T, than the red end t of the other Image p t from
the fame place MT which puts the Propofition paft di-
ipute. And this happens whether the third Prifm D H be
placed immediately after the two firft or at a great diftance
from them , fo that the Light refra(fled in the two firft
Prifms be either white and circular, or coloured and ob-
long when it falls on the third.

Exper. 6. In the middle of two thin Boards I made
round holes a third part of an Inch in Diameter, and in
the Window-fhut a much broader hole, being made to let
into my darkned Chamber a large beam of the Sun's
Light 5 I placed a Prifm behind the Shut in that beam to
relrad it towards the oppofite Wail, and clofe behind the
Prifm I fixed one of the Boards, in fuch manner that the
middle of the refrad:ed Light might pafs through the hole

made

[31]

made in it, and the reft be intercepted by the Board.
Then at the diftance of about twelve Feet from the firft
Board I fixed the other Board, in fuch manner that the
middle of the refra<5ted Light which came through the hole
in the firft Board and fell upon the oppofite Wall might
pafs through the hole in this other Board, and the reft be-
ing intercepted by the Board might paint upon it the co-
loured Spectrum of the Sun. And clofe behind this Board
I fixed another Prifm to refrad: the Light which came
through the hole. Then I returned fpeedily to the firft
Prifm, and by turning it flowly to and fi-o about its Axis,
I caufed the Image which fell upon the fecond Board to
move up and down upon that Board, that all its parts
might fucccflively pafs through the hole in that Board and
fall upon the Prifm behind it. And in the mean time, I
noted the places on the oppofite Wall to which that Light
after its Refraction in the fecond Prifm did pafs 5 and by
the difference of the places I found that the Light which
being moft refra(5led in the firft Prifm did go to the blew
end of the Image, was again more refracted in the fecond
Prifm than the Light which went to the red end of that
Image, which proves as well the firft Propofition as the
fecond. And this happened whether the Axis of the two
Prifms were parallel, or inclined to one another and to the
H®rizon in any given Angles.

Illuftration. Let r be the wide hole in the Window-fliut, p^v i g,
through which the Sun fhines upon the firft Prifm ABC,
and let the refraded Light fall upon the middle of the
Board D E, and the middle part of that Light upon the
hole G made in the middle of that Board. Let this tra-
jeded part of the Light fall again upon the middle of the
fecond Board d e and there paint fuch an oblong coloured
Image of the Sun as was defcribed in the third Experiment.

C32]

'By turning the Prifm ABC flowly to and fro about it^
Axis this Image will be made to move up and down the
Board d e, and by this means all its parts from one end to
the other may be made to pafs fucceflively through the
hole g which is made in the middle of that Board. In the
mean while another Prifm a b c is to be fixed next after
that hole^ to refracft the traje(5ted Light a fecond time.
And thefe things being thus ordered, I marked the places
M and N of the oppofice Wall upon which the refradled
Light fell,and found that whilft the two Boards and fecond
Prifm remained unmoved, thofe places by turning the firft
Prifm about its Axis were changed perpetually. For when
the lower part of the Light which fell upon the fecond
Board d e was call through the hole ^ it went to a lower
place M on the Wall , and when the higher part of that
Light was caft through the fame hole^, it went to a higher
place N on the Wall,- and wken any intermediate part of
the Light was caft through that hole it went to fome place
on the Wall between M and N. The unchanged Pontion
of the holes in the Boards, made the Incidence of the Rays
upon the fecond Prifm to be the fame in all cafes. And
yet in that common Incidence fome of the Rays were more
refracted and others lefs. And thofe were more refra(5ted
in this Prifm which by a greater Refraction in the firft
Prifm were more turned out of the way, and therefore for
their conftancy of being more refi:a6ted are defervedly cal-
led more Refrangible.

Exper. 7. At two holes made near one another in my
Window-fliut I placed two Prifms , one at each, which
might caft upon the oppofite Wall ( after the manner of
the third Experiment ) two oblong coloured Images of the
Sun. And at a little diftance from the Wall I placed a
long flender Paper with ftraight and parallel edges, and

ordered

C33l

ordered the Prifms and Paper fo, that the red Colour of
one Image might fall diredily upon one half of the Paper,
and the violet colour of the other Image upon the other
half of the fame Paper j fo that the Paper appeared of two
Colours , red and violet , much after the manner of the
painted Paper in the firft and fecond Experiments. Then
with a black Cloth I covered the Wall behind the Paper,
that no Light might be refleded from it to difturb the
Experiment, and viewing the Paper through a third Prifm
held parallel to it, I faw that half of it which was illumi-
nated by the Violet-light to be divided from the other
half by a greater Refradiion, elpecially when I went a good
way off from the Paper. For when I viewed it too near
at hand, the two halfs of the Paper did not appear fully
divided from one another , but feemed contiguous at one
of their Angles like the painted Paper in the firft Expe-
riment. Which alfo happened when the Paper was too
broad.

Sometimes inftead of the Paper I ufed a white Thred,
and this appeared through the Prifm divided into two Pa-
rallel Threds as is reprefented in the 19th Figure, where Fig^. ip.
D G denotes the Thred illuminated with violet Light
from D to E and with red Light from F to G, and d e fg
are the parts of the Thred feen by Refradion. If one half
of the Thred be conftantly illuminated with red, and the
other half be illuminated with all the Colours fuccefiively,
(which may be done by caufing one of the Prifms to be
turned about its Axis whilft the other remains unmoved)
this other half in viewing the Thred through the Prifm,
will appear in a continued right Line with the firft half
when illuminated with red , and begin to be a little divi-
ded from it when illuminated with Orange, and remove
further from it when illuminated with Yellow, and ftill

E further

[3+]

further when with Green, and further when with Blew, and
go yet further off when illuminated with Indigo, and fur-
theft when with deep Violet. Which plainly fhews, that
the Lights of feveral Colours are more and more Refran-
gible one than another, in this order of their Colours, Red,
Orange, Yellow, Green, Blew, Indigo, deep Violet 3 and
fo proves as M^ell the firft Propofition as the fecond.
pifT. 17. I caufed alfo the coloured Spe6trums PT and M N
made in a dark Chamber by the Refra6tions of two Prifms
to lye in a right Line end to end, as was defcribed above
in the fifth Experiment, and viewing them through a third
Prifm held Parallel to their length, they appeared no longer
in a right Line, but became broken from one another, as
they are reprefented 3.t p t and tn », the violet end m of the
Spectrum ?« n being by a greater Refradiion tranflated
further from its former place M T than the red end t of the
other Spedrum p t.
Fi<r. 20. I further caufed thofe two Spedlrums P T and MN to-
become co-incident in an inverted order of their Colours,
the red end of each falling on the violet end of the other,
as they are reprefented in the oblong Figure P T M N 5
and then viewing them through a Prifm D H held Paral-
lel to their length, they appeared not co-incident as when
viewed with the naked Eye , but in the form of two di-
ftincfl Spe6trums p t and m n eroding one another in the
middle after the manner of the letter X. Which fliews
that the red of the one Spe(5trum and violet of the other,
which were co-incident at P N and M T , being parted
from one another by a greater Refraction of the violet to
p and m than of the red to 71 and f, do differ in degrees of
Refrangibility.

I illuminated alfo a little circular piece of white Paper
all over witk the Lights of both Prilms intermixed, and

when

[35]

when it was illuminated with the red of one Spe(5lrum and
deep violet of the other , fo as by the mixture of thoic
Colours to appear all over purple , I viewed the Paper,
firfi; at a lefs diflance , and then at a greater , through a
third Prifm 5 and as I went from the Paper, the refra(5ted
Image thereof became more and more divided by the un-
equal Refra6tion of the two mixed Colours, and at length
parted into two diftind: Images, a red one and a violet one,
whereof the violet was furtheft from the Paper, and there-
fore fufFered the greatefl: Refracflion. And when that Prifm
at the Window which caft the violet on the Paper was ta-
ken away,the violet Image difippeared j but when the other
Prifm was taken away the red vaniflied : which fiiews that
thefe two Images were nothing elfe than the Lights of the
two Prifms which had been intermixed on the purple Pa-
per, but were parted again by their unequal Refracflions
made in the third Prifm through which the Paper was
viewed. This alfo was oblervable that if one of the
Prifms at the Window, fuppofe that which caft the violet
on the Paper, was turned about its Axis to make all the
Colours in this order, Violet, Indigo, Blew, Green, Yel-
low, Orange, Red, fall fucceffively on the Paper from that
Prifm, the violet Image changed Colour accordingly, and
in changing Colour came nearer to the red one, until when
it was alfo red they both became fully co-incident.

I placed alfo two paper circles very near one another,
the one in the red Light of one Prifm, and the other in
the violet Light of the other. The circles were each of
them an Inch in Diameter, and behind them the Wall was
dark that the Experiment might not be difturbed by any
Light coming from thence. Thefe circles thus illuminated,
I viewed through a Prifm fo held that the Refraction might
be made towards the red circle , and as I went from them

-E 2 they

they came nearer and nearer together, and at length be-
came co-incident 3 and afterwards when I went ftill further
off, they parted again in a contrary order, the violet by a
greater Refrad:ion being carried beyond the red.

Exper. 8. In Summer when the Sun's Light ufes to
be ftrongeft, I placed a Prifm at the hole of the Window-
fhut, as in the third Experiment, yet fo that its Axis might
be Parallel to the Axis of the World, and at the oppofite
Wall in the Sun s refracted Light, I placed an open Book.
Then going Six Feet and two Inches from the Book, I
placed there the abovementioned Lens,by which the Light
refle*5led from the Book might be made to converge and
meet again at the diftance of fix Feet and two Inches be-
hind the Lens , and there paint the Species of the Book
upon a flieet of white Paper much after the manner of the
fecond Experiment. The Book and Lens being made fall,
I noted the place where the Paper was, when the Letters
of the Book, illuminated by the fullefl red Light of the
Solar Image falling upon it, did caft their Species on that-
Paper moft diftindly 5 And then I ftay'd till by the Mo-
tion of the Sun and confecjuent Motion of his Image on
the Book, all the Colours from that red to the middle of
the blew pafs'd over thofe Letters 3 and when thofe Letters
were illuminated by that blew, I noted again the place of
the Paper when they caft their Species moft diftindily upon
it : And I found that this laft place of the Paper was nearer
to the Lens than its former place by about two Inches and
an half, or two and three quarters. So much fooner there-
fore did the Light in the violet end of the Image by a grea-
ter Refradlion converge and meet , than the Light in the
red end. But in trying this the Chamber was as dark as I
could make it. For if thefe Colours be diluted and weak-
ned by the mixture of any adventitious Light, the diftance

between

[37]

between the places of the Paper will not be fo great. This
diftance in the fecond Experiment where the Colours of
natural Bodies were made ufe of, was but an Inch and a
half, by reafon of the imperfe(5tion of thofe Colours. Here
in the Colours of the Prifm , which are manifeftly more
full, intenfe, and lively than thofe of natural Bodies, the
diftance is two Inches and three quarters. And were the
Colours ftill more full , I queftion not but that the di-
ftance would be confiderably greater. For the coloured
Light of the Prifm, by the interfering of the Circles de-
fcribed in the 1 1 th Figure of the fifth Experiment, and alfo
by the Light of the very bright Clouds next the Sun's
Body intermixing with thefe Colours, and by the Light
fcattered by the inequalities in the polifh of the Prifm, was
fo very much compounded, that the Species which thofe
faint and dark Colours, the Indigo and Violet, caft upon
the Paper were not diftin6t enough to be well obferved.

Expcr. 9. A Prifm, whofe two Angles at its Bafe were
equal to one another and half right ones, and the third
a right one, I placed in a beam of the Sun's Light let in-
to a dark Chamber through a hole in the Window-fhut
as in the third Experiment. And turning the Prifm flowly
about its Axis until all the Light which went through one
of its Angles and was refracted by it began to be refle<5led
by its Bafe , at which till then it went out of the Glafs,
I obferved that thofe Rays which had fuffered the greateil
Refraction were fooner reflected than the reft. I conceived
therefore that thofe Rays of the reflected Light, which
were moft Refrangible, did firft of all by a total Reflexion
become more copious in that L'ght than the reft , and
that afterwards the reft alfo, by a total Reflexion, be-
came as copious as thefe. To try this , I made the re-
fleded Light pafs through another Prifm, and being refra-

aed

[38]

ded by it to fall afterwards upon a fbeet of white Paper
placed at fome diftance behind it, and there by that Re-
ira<5lion to paint the ufual Colours of the Prifm. And
then caufing the firft Prifm to be turned about its Axis as
above, I obferved that when thofeRays which in this Prifm
had fuffered the greateft Refrad:ion and appeared of a blew
and violet Colour began to be totally refleded , the blew
and violet Light on the Paper which was mofl refra6ted
in the fecond Prifm received a fenfible increafe above that
■of the red and yellow, which was leaft refracSted 5 and
afterwards when the reft of the Light which was green,
yellow and red began to be totally reflected in the firft
Prifm, the hght ofthofe Colours on the Paper received as
great an increafe as the violet and blew had done before.
Whence 'tis manifeft, that the beam of Light refledied by
the Bafe of the Prifm, being augmented firft by the more
Refrangible Rays and afterwards by the lefs Refrangible
ones, is compounded of Rays differently Refrangible.
And that all fuch refledled Light is of the fame Nature
with the Sun's Light, before its Incidence on the Bafe of
the Prifm, no Man ever doubted : it being generally al-
lowed, that Light by fuch Reflexions fuffers no Alteration
in its Modifications and Properties. I do not here take
notice of any Refrad:ions made in the Sides of the firft
Prifm, becaufe the Light enters it perpendicularly at the
firft Side, and goes out perpendicularly at the fecond Side,
and therefore fuffers none. So then, the Sun's incident
Light being of the fame temper and conftitution with his
emergent Light, and the laft being compounded of Rays
differently Refrangible , the firft muft be in like manner
compounded.
Fig. 1 1 . Illujlration. In the 2 1 th Figure, A B C is the firft Prifm,
B C its Bafe, B and C its equal Angles at the Bafe, each

[39]

of 45 degrees, A its Re6langular Vertex, F M a beam of
the Sun's Light let into a dark Room through a hole F
one third part of an Inch broad, M its Incidence on theBafe
of the Prifm,M G a lefs refraded Ray, M H a more refradt-
ed Ray, M N the beam of Light refle(5led from the Bafe ,
V X Y the fecond Prifm by which this beam in paffing
through it is refrad:ed, N t the lefs refracted Light of this
beam, and N p the more refraded part thereof When the
firft Prifm A B C is turned about its Axis according to the
order of the Letters ABC, the Rays M H emerge more
and more obliquely out of that Prifm, and at length after
their mofl: oblique Emergence are refled:ed towards N,
and going on to p do increafe the number of the Rays N p.
Afterwards by continuing the motion of the firft Prifm, the
Rays MG are alfo refled:ed to N and increale the number of
the Rays N t. And therefore the Light M N admits into
its Compoficion, firft the more Refrangible Rays, and then,
the lefs Refrangible Rays, and yet after this Compofition
is of the f\me Nature with the Sun's immediate Light F M,
the Reflexion of the fpecular Bafe B C caufing no Altera-
tion therein.

Exper. 1 o. Two Prifms, which were alike in fhape, I
tied fo together, that their Axes and oppofite Sides being
Parallel, they compofed a Parallelopiped. And, the Sun
fhining into my dark Chamber through a little hole in the
Window-fhut, I placed that Parallelopiped in his beam at
fome diftance from the hole, in fuch a pofture that the Axes
of the Prifms might be perpendicular to the incident Rays,
and that thofe Rays being incident upon the firft Side of
one Prifm, might go on through the two contiguous Sides
of both Prifms, and emerge out of the laft Side of the fe-
eond Prifm. This Side being Parallel to the firft Side of
the firft Prifm , caufed the emerging Light to be Parallel

[4°]

CO the Incident. Then, beyond thefe two Prifms I placed
a third, which might refrad: that emergent Light, and by
that Refrad:ion caft the ufual Colours of the Prifm upon
the oppofite Wall, or upon a fheet of white Paper held at
a convenient diftance behind the Prifm for that refraded
Light to fall upon it. After this I turned the Parallelopiped
about its Axis, and found that when the contiguous Sides
of the two Priims became fo oblique to the incident Rays
that thofe Rays began all of them to be refled:ed , thofe
Rays which in the third Prifm had fuflfered the greateft Re-
fraction and painted the Paper with violet and blew, were
firft of all by a total Reflexion taken out of the tranfmitted
Licrht, the reft remaining and on the Paper painting their
Colours of Green, Yellow, Orange, and Red as before 5
. and afterwards by continuing the motion of the two Prifms,
the reft of the Rays alfo by a total Reflexion vanifhed in
. order, according to their degrees of Refrangibility. The
Li^ht therefore which emerged out of the two Prifms is
compounded of Rays differently Refrangible , feeing the
more Refrangible Rays may be taken out of it while the
lefs Refrangible remain. But this Light being trajeded
only through the Parallel Superficies of the two Prifms, if
it fuffered any change by the Refraction of one Superficies
it loft that impreflion by the contrary Refrad;ion of the
other Superficies, and fo being reftored to its priftine con-
ftitution became of the fame nature and condition as at firft
before its Incidence on thofe Prifms 3 and therefore, before
its Incidence, was as much compounded of Rays differently
Refrangible as afterwards.
Fig. 11. lllufiration. In the iith Figure ABC and B C D are the
the two Prifms tied together in the form of a Parallelo-
piped, their Sides BC and CB being contiguous, and
their Sides A B and C D Parallel. And H J K is the third

Prifm,

[41]

Prifm, by which the Sun's Light propagated through the
hole F into the dark Chamber, and there pafling through
thofe fides of the Prifms AB, BC, CB and CD, is refra-
(fted at O to the white Paper PT, falling there partly upon
P by a greater Refradiion, partly upon T by a lefs Refra-
d:ion, and partly upon R and other intermediate places by
intermediate Refractions. By turning the Parallelopiped
ACBD about its Axis, according to the order of the Let-
ters A,C,D,B, at length when the contiguous Planes BC
and CB become fufficiently oblique to the Rays F M,
which are incident upon them at M, there will vaniCh to-
tally out of the refradled Light OPT, firft of all the moll
refraded Rays OP, (the reft OR and OT remaining as
before) then the Rays O R and other intermediate ones,
and laftly, the leaft refraded Rays O T. For when the
Plane B C becomes fufficiently oblique to the Rays inci-
dent upon it, thofe Rays will begin to be totally refled;-
ed by it towards N 3 and firft the moft Refrangible Rays
will be totally reflected (as was explained in the preceding
experiment) and by confequence muft firft difappear at P,
and afterwards the reft as they are in order totally refled:-
ed to N, they muft difappear in the fame order at R and
T. So then the Rays which at O fuffer the greateft Re-
fraction, may be taken out of the Light MO whilft the reft
of the Rays remain in it, and therefore that Light MO is
Compounded of Rays differently Refrangible. And be-
caufe the Planes A B and C D are parallel, and therefore
by equal and contrary Refractions deftroy one anothers
Effects, the incident Light F M muft be of the fame kind
and nature with the emergent Light M O, and therefore
doth alfo confift of Rays difl^erently Refrangible. Thefe
two Lights FM and MO, before the moft relrangible Rays
are feparated out of the emergent Light MO agree inCo-

F lour,

lour, and in all other properties fo far as uny obfervadion-
reaches, and therefore are defervedly reputed of the fanrve
Nature and Conftitution, and by confequence rhe one is
compounded as well as the other. But after the moft Re-
frangible Rays begin to be totally refleifled, and thereby
feparated out of the emergentLightMO,that Light changes
its Colour from white to a dilute and faint yellow, a pretty
good orange, a very full red fucceflively and then totally
vaniflies. For after the moft Refrangible Rays which paint
the Paper at P with a Purple Colour, are by a total re-
flexion taken out of the Beam of light M O, the reft of
the Colours which appear on the Paper at R and T being
mixed in the light M O compound there a faint yellow,
and after the blue and part of the green which appear on
the Paper between P and R are taken away, the reft which
appear between R and T (that is the Yellow, Orange, Red
and a little Green) being mixed in the Beam M O com-
pound there an Orange 3 and when all the Rays are by re-
flexion taken out of the Beam MO, except the leaft Refran-
gible, which at T appear of a full Red, their Colour is
the fame in that Beam M O as afterwards at T, the Re-
fraction of the Prifm HJK ferving only to feparate the
differently Refrangible Rays, without making any alteration
in their Colours, as fliall be more fully proved hereafter.
All which confirms as well the firft Propofition as the fe-
c&ftd.

Scholium. If this Experiment and the former be conjoyned
J*/^. 22. and made one, by applying a fourth Prifm VXY to re-
fract the refled:ed Beam M N towards tp^ the conclufion
will be clearer. For then the light N/> which in rhe 4th
Prifm is more refradled, will become fuller and ftronger
when the Light O P, which in the third Prifm H J K is
more refracted, vaniflies at P j and afterwards when th« lefs

refracted

[43]

refracted Light O T vaniflies at T,the lefs refraded Light
Nf will become encreafed whilft the more refradled Light
at p receives no further encreafe. And as the traje6ted
Beam M O in vanifhing is always of fuch a Colour as
ought to refult from the mixture of the Colours which
fall upon the Paper PT, fo is the refledied Beam MN al-
ways of fuch a Colour as ought to refult from the mix-
ture of the Colours which fall upon the Paper p t. For
when the mofl refrangible Rays are by a total Reflexion
taken out of the Beam M O, and leave that Beam of an
Orange Colour, the excefs of thofe Rays in the refle6te<i
Light, does not only make the Violet, Indigo and Blue at
p more full, but alfo makes the Beam M N change from
the yellowiili Colour of the Sun's Light, to a pale white in-
clining to blue, and afterward recover its yellowifli Co-
lour again, fo foon as all the reft of the tranfmitted light
MOT is reflefted.

Now feeing that in all this variety of Experiments,
whether the trial be made inLight reflected, and that either
from natural Bodies, as in the firft and fecond Experiment,
or Specular, as in the Ninth 5 or in Light refrad:ed, and
that either before the unequally refradled Rays are by di-
verging feparated from one another, and lofing their white-
nefs which they have altogether, appear feverally of feve-
ral Colours, as in the fifth Experiment j or after they are
feparated from one another, and appear Coloured as in the
fixth, feventh, and eighth Experiments 3 or in Light tra-
jeded through Parallel fuperficies, deftroying each others
Effeds as in the 1 oth Experiment 5 there are always found
Rays, which at equal Incidences on the fame Medium fuf-
fer unequal Refrad:ions, and that without any fplitting or
dilating of fingle Rays, or contingence in the inequality
of the Refradions, as is proved in the fifth and fixth Ex-

F 2 periments;

[44]

periments^ and feeing the Rays which differ in Refrangibi-
iiry may be parted and forced from one another, and that
cither by Refradion as in the third Experiment, or by Re-
flexion as in the tenth, and then the feveral forts apart at
equal Incidences fuffer unequal Refradtions, and thole forts
are more refracted than others after feparation, which were
more refracfled before it, as in the fixth and following Ex-
periments, and if the Sun's Light be trajed:ed through three
or more crofs Prifms fucceflively, thofe Rays which in the
firfl Prifm are refraded more than others are in all the fol-
lowing Prifms, refradied more then others in the fame rate
and proportion, as appears by the fifth Experiment 3 it's
manifeft that the Sun's Light is an Heterogeneous mixture of
Rays, fome of which are conftantly more Refrangible then
others, as was to be propofed.

PROP. III. Theor. III.

The Su?is Light conjifls of (^ys dijfenng in ^flexibility., and
thofe ^ys are more ^flexible thati others which are more (?(f-
frangible.

THIS is manifeft by the ninth and tenth Experi-
ments : For in the ninth Experiment, by turning
the Prifm about its Axis, until the Rays within it which in
going out into the Air were refracted by its Bafe, became
10 oblique to that Bafe, as to begin to be totally refle<5ted
thereby 3 thofe Rays became firft of all totally refle<5ted,
which before at equal Incidences with the reft had fuffered
the greateft Refra(5tion. And the fame thing happens in
the Reflexion made by the common Bafe of the two Prifms
in the tenth Experiment.

[45 J

PROP. IV. Prob. I.

To /eparate from one another the Heterogeneous ^ys of

Compound Light.

THE Heterogeneous Rays are in fomt meafure fepa-
rated from one another by the Refra6tion of the
Prifm in the third Experiment, and in the fifth Experiment
by taking away the Penumbra from the RediHnear fides of
the Coloured Image, that feparation in thofe very Rectili-
near fides or ftraight edges of the Image becomes perfect.
But in all places between thofe rectilinear edges, thofe in^-
numerable Circles there defcribed, which are feverally illu-
minated by Homogeneral Rays, by interfering with one
another, and being every where commixt, do render the
Light fuilficiently Compound. But if thefe Circles, whilft
their Centers keep their diftances and pofitions, could be
made lefs in Diameter, their interfering one with another
and by confequence the mixture of the Heterogeneous
Rays would be proportionally diminiflied. In the 2 3thF^. 23.
Figure let AG, B H, C J, D K, EL, F M be the Circles
which fo many forts of Rays flowing from the fameDifque
of the Sun, do in the third Experiment illuminate 5 of all
which and innumerable other intermediate ones lying in a
continual Series between the two Re(5tilinear and Parallel
edges of the Sun's oblong Image P T, that Image is com-
pofed as was explained in the fifth Experiment. And lee
4^, bh, cij dl{^ el J fm be fo many lefs Circles lying in
a like continual Series between two Parallel right Lines af
and g m with the fame diftances between their Centers,
and illuminated by the fame forts of Rays, that is the
Circle ag with the fame fore by which the correfponding

Circle

Circle AG was illuminated, and the Circle hh with the fame
fort by which the correfpondingCircle BHwas illuminated,
and the reft of the Circles c *', dk, elj fm refpedively,
with the fame fores of Rays by which the feveral corre-
fponding Circles C J, D K, EL, FM were illuminated.
In the Figure P T compofed of the greater Circles, three
of thofe Circles AG, B H, CJ, are fo expanded into one
another, that the three forts of Rays by which thofe Cir-
cles are illuminated, together with other innumerable forts
of intermediate Rays, are mixed at Q.R in the middle of
the Circle B H. And the like mixture happens through-
out almoft the whole length of the Figure P T. But in
the Figure p t compofed of the lefs Circles, the three lefs
Circles ag^ b h, c /, which anfwer to thofe three greater, do
not extend into one another 5 nor are there any Vv^here
mingled fo much as any two of the three forts of Rays
by which thofe Circles are illuminated, and which in the
Figure P T are all of them intermingled at B H.

Now he that fliall thus confider it, will eafily underftand
that the mixture is diminifhed in the fame Proportion
with the Diameters of the Circles. If the Diameters of
the Circles whilft their Centers remain the fame, be made
three times lefs than before, the mixture will be alfo three
times lefs 5 if ten times lefs, the mixture will be ten times
lefs, and fo of other Proportions. That is, the mixture
of the Rays in the greater Figure P T will be to their mix-
ture in the lefs p t, as the Latitude of the greater Figure is
to the Latitude of the lefs. For the Latitudes of thefe Fi-
gures are equal to the Diameters of their Circles. And
hence it eafily follows, that the mixture of the Rays in the
refracted Spectrum pt is to the mixture of the Rays in the
dire<5t and immediate Light of the Sun, as the breadth of
that Spedirum is to the difference between the length and
breadth of the fame Spe^^rum. So

C47l

Se^ then, if we woul<J diminifli eli^ m'ixtmte of the Rays,
we af€ to d'immifb the Diamerers &§ the Cireres. Now
thefe wouW be diminiflied if the Sun^s Diameter to which
they anfwer could be made lefs than it is, or (which comes
to the fame puipofe) if without Dgofs, at a great diftance
from the Prifm towards the Sun, fome opake body were
placed, with a round hole in the middle of it, to intercept
all the Sun's Light, excepting fo much as coming from
the middle of his Body could pafs through that hole to
the Prifm. For fo the Circles A G, B H and the reft,
would not any longer anfwer to the whole Difque of the
Sun , but only to that part of it which could be feen
from the Prifm through that hole, that is to the apparent
magnitude of that hole viewed from the Prifm. But that
thefe Circles may anfwer more diftin<5lly to that hole a
Lens is to be placed by the Prifm to caft the Image of the
hole, (that is, every one of the Circles A G, B H, <6"c.) di-
ftindly upon the Paper at P T, after fuch a manner as by
a Lens placed at a Window the Species of Objedls abroad
are caft diftindly upon a Paper within the Room, and the
Rectilinear Sides of the oblong folar Image in the fifth
Experiment became diftind: without any Penumbra. If
this be done it will not be neceflary to place that hole
very far off, no not beyond the Window. And therefore
inftead of that hole, I ufed the hole in the Window-fliut
as follows.

Exper. 1 1 . In the Sun's Light let into my darkned
Chamber through a fmall round hole in my Window-
fliut, at about i o or 1 1 Feet from the Window, I placed
a Lens , by which the Image of the hole might be di-
ftinClly caft upon a fheet of white Paper, placed at the
diftance of fix, eight, ten or twelve Feet from the Lens.
For according to the difference of the Lenfes I ufed various

diftances,

[48]

diftances , which I think not worth the while to defcribe.
Then immediately after the Lens I placed a Prifm, by
which the trajeded Light might be refracted either up-
wards or Tideways, and thereby the round Image which
the Lens alone did caft upon the Paper might be drawn
out into a long one with Parallel Sides , as in the third
Experiment. This oblong Image I let fall upon another
Paper at about the fame diftance from the Prifm as be-
fore, moving the Paper either towards the Prifm or from
it, until I found the juft diftance where the Rectilinear
Sides of the Image became moft diftin(5l. For in this cafe
the circular Images of the hole which compofe that Image
after the fame manner that the Circles d^, bh, ci, &c. do

Pig, 25. the Figure p f , were terminated moft diftin<5tly without any
Penumbra, and therefore extended into one another the
leaft that they could, and by confequence the mixture of
the Heterogeneous Rays was now the leaft of all. By this

fin 1 ^ , means I ufed to form an oblong Image (fuch as is p t) of
afid 24. circular Images of the hole (fuch as are a^, bh, ci^ &c. )
and by ufing a greater or lefs hole in the Window-fhut, I
made the circular Images ag, b A, c /, &c. of which it was
formed, to become greater or lefs at pleafure, and thereby
the mixture of the Rays in the Image pt to be as much
or as little as I defired.

Fk. 24. Illujlrat'wn. In the 24th Figure, F reprefents the circular
hole in the Window-ftiut, M N the Lens whereby the
Image or Species of that hole is caft diftin6tly upon a
Paper at J, ABC the Prifm whereby the Rays are at their
emerging out of the Lens refracted from J towards ano-
ther Paper at p t, and the round Image at J is turned into
an oblong Image p t falling on that other Paper. This
Image p t confifts of Circles placed one after another in a
Redilinear order, as was fufficiently explained in the fifth

Experiment 3

[49]

Experiment • and thefe Circles are equal to th£ Circle I,
and confequently anfwer in Magnitude to the hole F ; and
therefore by diminifhing that hole they may be at pleafure
diminifhed , whirft their Centers remain in their places.
By this means I made the breadth of the Image pt to be
forty times, and fometimes lixty or feventy times lefs than
its length. As for inftance, if the breadth of the hole F
be - of an Inch, and MF the diftance of the Lens from
the hole be i 2 Feet 3 and if /? B or pM the diftance of
the Image pt from the Prifm or Lens be 10 Feet, and the
refrading Angle of the Prifm be 6i degrees, the breadth
of the Image p t will be ~ of an Inch and the length about
fix Inches, and therefore the length to the breadth as 71
to 1, and by confequence the Light of this Image 71 times
lefs compound than the Sun's direA Light. And Light
thus far Simple and Homogeneal, is fufficient for trying
all the Experiments in this Book about fimple Light. For
the compofition of Heterogeneal Rays is in this Light fo
little that it is fcarce to be difcovered and perceived by
fenfe, except perhaps in the Indigo and Violet j for thefe
being dark Colours, do eafily funer a fenfible allay by that
little fcattering Light which ufes to be refradled irregularly
by the inequaliteis of the Prifm.

Yet inftead of the circular hole F, 'tis better to fubfti-
tute an oblong hole fliaped like a long Parallelogram
with its length Parallel to the Prifm ABC. For if this
hole be an Inch or two long, and but a tenth or twentieth
part of an Inch broad or narrower : the Light of the Image
p t will be as Simple as before or fimpler, and the Image
will become much broader, and therefore more fit to have
Experiments tried in its Light than before.

Inftead of this Parallelogram-hole may be fubftitnted a
Triangular one of equal Sides, whofe Bafe for inftance is

G about

C50]

about the tenth part of an Inch, and its height an Inch of
more. For by this means , if the Axis of the Prifm be
Parallel to the Perpendicular of the Triangle , the Image
Fig. 1'^. pt will now be formed of Ec^uicrural Triangles ag, hhj ci,
^k.-) ^h f'^^y ^^- ^^^ innumerable other intermediate ones
anfwering to the Triangular hole in fhape and bignefs,and
lying one after another in a continual Series between two
Parallel Lines af z.nAgm. Thefe Triangles are a Httle
intermingled at their Bafes but not at their Vertices, and
therefore the Light on the brighter fide af of the Image
where the Bafes of the Triangles are is a little compounded,
but on the darker fide^?w is altogether uncompounded,
and in all places between the fides the Compofition is
Proportional to the diftances of the places from that ob-
fcurer fide^ m. And having a Spedlrum p t o^ fuch a
Compofition, we may try Experiments either in its ftronger
and lefs fimple Light near the fide af, or in its weaker
and fimpler Light near the other fide / w, as it (hall feem
moft convenient.

But in making Experiments of this kind the Chamber
ought to be made as dark as can be, leaft any forreign
Light mingle it felf with the Light of the Spedrum p t,
and render it compound 5 efpecially if we would try Ex-
periments in the more fimple Light next the fide g ??i of
the Spectrum 5 which being fainter, will have a lefs Pro-
portion to the forreign Light, and fo by the mixture of
that Light be more troubled and made more compound.
The Lens alfo ought to be good, fuch as may ferve for
Optical Mksy and the Prifm ought to have a large Angle,
fuppofe of 7© degrees, and to be well wrought, being
made of Glafs free from Bubbles and Veins, with its fides
not a little Convex or Concave as ufually happens but
truly Plane,and its pollifli elaborate, as in working Optick-

glafles

C5I]

glafles , and not fuch as is ufiially wrought with Putty,
whereby the edges of the Sand-holes being worn away,
there are left all over the Glafs a numberlefs company of
very little Convex polite rifings like Waves. The edges
alfo of the Prifm and Lens fo far as they may make any
irregular Refraction, muft be covered with a black Paper
glewed on. And all the Light of the Sun's beam let into
the Chamber which is ufelefs and unprofitable to the Ex-
periment, ought to be intercepted with black Paper or other
black Obftacles. For otherwife the ufelefs Light being
refled;ed every way in the Chamber , will mix with the
oblong Spectrum and help to difturb it. In trying thefe
things fo much Diligence is not altogether neceflary, but
it will promote the fuccefs of the Experiments, and by a
very fcrupulous Examiner of things deferves to be applied.
It's difficult to get glafs Prifms fit for this purpofe, and
and therefore I ufed fometimes Prifmatick Veffels made
with pieces of broken Looking- glaffes, and filled with rain
Water. And to increafe the Refradlion, I fometimes im-
pregnated the Water ftrongly with Saccharum Saturni.

PROP. v. Theor. IV.

Homogeneal Light is re/rafted regularly ivithout any Dilatation

f putting or Jhattering of the ^ys , and the confufed Vijlon

of Objefis feen through ^fraHing 'Bodies by Hetcrogeneal

Light arifes from the different ^efrangibility of JeVeral forts

of llays.

TH E firfl: Part of this Propofition has been already
fufficiently proved in the fifth Experiment, and will
further appear by the Experiments which follow.

G 2 Exper. \ i .

[52]

Exper. 12. In the middle of a black Paper I made i
round hole about a fifch or fixth part of an Inch in Dia-
meter. Upon this Paper I caufcd the Spedrum of Homo-
geneal Light defcribed in the former Propofition , fo to
fall, that lome part of the Light might pafs through the
hole of the Paper. This tranfmicted part of the Light I
refracted with a Prifm placed behind the Paper, and let-
ting this refracfled Light fall perpendicularly upon a white
Paper two or three Feet diftant from the Prifm, I foand
that the Sped:rum formed on the Paper by this Light was
not oblong, as when 'tis made (in the third Experiment)
by Refracting the Sun's compound Light, but was (fo far
as I could judge by my Eye) perfed:ly circular, the length
being no greater than the breadth. Which fhews that this
Light is refraCled regularly without any Dilatation of the
Rays.

Exper. 1 ^ . In the Homogeneal Light I placed a Circle
of-' of an Inch in Diameter, and in the Sun's unrefrad:cd
Heterogeneal white Light I placed another Paper Circle of
the fame bignefs. And going from the Papers to the diflance
of fomeFeet, I viewed both Circles through a Prifm. The
Circle illuminated by the Sun's Heterogeneal Light appear-
ed very oblong as in the fourth Experiment , the length
being many times greater than the breadth : but the other
Circle illuminated with Homogeneal Light appeared Cir-
cular and difl;ind:ly defined as when 'tis viewed with the
naked Eye. Which proves the whole Propofition.

Exper. 1 4. In the Homogeneal Light I placed Flies and
fuch like Minute Objeds, and viewing them through a
Prifm , I faw their Parts as diftindly defined as if I had
■viewed them with the naked Eye. The fame Objeds pla-
ced in the Sun's unrefradied Heterogeneal Light which was
white I viewed alfo through a Prifm, and faw them moft

confufedly

C53]

confufedly defined, fo thatlcould not diftinguifii their fmal*
* [er Parts from one another. I placed alfo the Letters of a
fmall Print one while in the Homogeneal Light and then
in the Heterogeneal, and viewing them through a Prifm,
they appeared in the latter cafe fo confufed and indiftinfib
that I could not read them 5 but in the former they ap-
peared fo diftind: that I could read readily, and thought
I [aw them as diftinit as when I viewed them with my
naked Eye. In both cafes I viewed the fame Objed:s
through the fame Prifm at the fame diftance from me and
in the fame Situation. There was no difference but in the
Light by which the Objects were illuminated , and which
in one cafe was Simple and in the other Compound, and
therefore the diftind Vifion in the former cafe and confu-
fed in the latter could arife from nothing elfe than from
that difference of the Lights. Which proves the whole
Propoficion.

And in thefe three Experiments it is further very remar-
kable, that the Colour of Homogeneal Light was never
changed by the Refrad;ion»

PROP. VI. Theor. V.

TT^e Sine of Incidence of e^ery ^ay confldered apart ^ is to its Sine
of ^efraHion in a p)} en ^tio.

THAT every Ray confidered apart is conftant to
it felf in fome certain degree of Refrangibility, is
fufficiently manifeft out of what has been faid. Thofe
Rays which in the firft Refradion are at equal Incidences
moft refraded, are alfo in the following Refradions at
equal Incidences moft refraded j and fo of the leaft Re-
frangible, and the reft which have any mean degree of

Refran-

[54]

Refrangibilicy, as is manifefl by the 5tk, 6th, 7th, 8th,
and 9th Experiments. And thofe which the firfl: time at •
like Incidences are equally refracted, are again at like In-
cidences equally and uniformly refracted, and that whe-
ther they be refradled before they be feparated from one
another as in the 5 th Experiment, or whether they be re-
fracted apart, as in the i 2 th, i ^rh and 14th Experiments.
The Refraftion therefore of every Ray apart is regular,
and what Rule that Refradion obferves we are now
to fliew.

Th« late Writers in Opticks teach, that the Sines of In-
cidence are in a given Proportion to the Sines of Refra-
d:ion, as was explained in the 5th Axiom 5 and fome by
Inftruments fitted for meafuring Refradions, or otherwife
experimentally examining this Proportion, do acquaint us
that they have found it accurate. But whilft they, not
underftanding the different Refrangibility of feveral Rays,
conceived them all to be refra6ted according to one and
the fame Proportion, 'tis to be prefumed that they adapted
their Meafures only to the middle of the refracted Light 3
fo that from their Meafures we may conclude only that
the Rays which have a mean degree of Refrangibilicy ,
that is thofe which when feparated from the reft appear
green, are refracted according to a given Proportion of
their Sines. And therefore we are now to fhew that the
like given Proportions obtain in all the reft. That it
fhould be fo is very reafonable, Nature being ever confor-
mable to her felf : but an experimental Proof is defired.
And fuch a Proof will be had if we can fliew that the
Sines of Refraction of Rays differently Refrangible are
one to another in a given Proportion when their Sines of
Incidence are equal. For if the Sines of Refraction of all
the Rays are in given Proportions to the Sine of Refraction

L55]

of a Ray which has a mean degree of Refrangibility, and
this Sine is in a given Proportion to the equal Sines of
Incidence, thofe other Sines of Refraction will alfo be in
given Proportions to the equal Sines of Incidence. Now
when the Sines of Incidence are equal , it will appear by
the following Experiment that the Sines of Refraction are
in a given Proportion to one another.

Exper. 1 5 . The Sun fliining into a dark Chamber
through a little round hole in the Window- fhut, let S re-Pi^- ^<^»
prefent his round white Image painted on the oppofite
Wall by his dire6t Light, P T his oblong coloured Image
made by refracting that Light with a Prifm placed at the
Windowj and pt, or ip it^ or ^p 3 f, hisoblong coloured
Image made by refraCting again the fame Light fideways
with a fecond Prifm placed immediately after the firft in
a crofs Pofition to it, as was explained in the fifth Experi-
ment : that is to fay, pt when the RefraCtion of the fecond
Prifm is fraall, ip it when its RefraCtion is greater, and
^p T,t when it is greateft. For fuch will be the diverfity
of the Refractions if the refraCting Angle of the fecond
Prifm be of various Magnitudes 5 fuppofe oF fifteen or
twenty degrees to make the Image p ?, of thirty or
forty to make the Image ip 1 1^ and of fixty to make
the Image 3 /' 3 f . But for want of folid Glafs Prifms with
Angles of convenient bignefles, there may be Veflels
made of polifhed Plates of Glafs cemented together in the
form of Prifms. and filled with Water. Thefc things being
thus ordered, I obferved that all the folar Images or co-
loured SpeCtrums V T, pt, ip it, 3;? 3 f did very nearly
converge to the place S on which the direCt Light of the
Sun fell and painted his white round Image when the
Prifms were taken away. The Axis of the SpeCtrum PT,
that is the Line drawn through the middle of it Parallel to

its

its Redilinear Sides, did when produced pafs exadly through
the middle of that white round Image S. And when the
Refraction of the fecond Prifm was equal to the Refradlion
of the firfl, the refra6ling Angles of them both being about
60 degrees, the Axis of the Spectrum ^/^ ^ f made by that
Refradion, did when produced pafs alfo through the mid-
dle of the fame white round Image S. But when the Re-
fraction of the fecond Prifm was lefs than that of the firft,
the produced Axes of the Spedrums tp or it ip made
by that Refra<ftion did cut the produced Axis of the Spe-
ctrum TP in the Points w and «, a little beyond the Cen-
ter of that white round Image S. Whence the Proportion
of the Line ^ f T to the Line 3/? P was a little greater than
the Proportion of 2 tT to i^P, and this Proportion a little
greater than that of tT top]?. Now when the Light of
the Spectrum P T falls perpendicularly upon the Wall, thofe
Lines ^fT, ^p^, and it T, i/^P and t'T,/?P,are the Tan-
gents of the Refractions 5 and therefore by this Experiment
the Proportions of the Tangents of the RefraCtions are ob-
tained, from whence the Proportions of the Sines being deriv-
ed, they come out equal, fo far as by viewing the SpeCtrums
and ufing fome Mathematical reafoning I could Eftimate.
For I did not make an Accurate Computation. So then
the Propofition holds true in every Ray apart, fo far as ap-
pears by Experiment. And that it is accurately true may
be demonftrated upon this Suppofition, Tl^at 'Bodies refraEl
Light by ciEling upon its ^ys in Lines Perpendicular to their
Surfaces. But in order to this Demonftration, I muft di-
ftinguifh the Motion of every Ray into two Motions, the
one Perpendicular to the refraCting Surface, the other Pa-
rallel to it, and concerning the Perpendicular Motion lay
down the following Propolltion.

C57]

If any Motion or moving thing whatfoever be incident
with any velocity on any broad and thin Space termina-
ted on both fides by two Parallel Planes, and in its paflage
through that fpace be urged perpendicularly towards the
further Plane by any force which at given diflances from
the Plane is of given quantities 3 the perpendicular Velo-
city of that Motion or Thing, at its emerging out of that
fpace, fhall be always equal to the Square Root of the
Summ of the Square of the perpendicular Velocity of
that Motion or Thing at its Incidence on that fpace ;
and of the Square of the perpendicular Velocity which
tKat Motion or Thing would have at its Emergence, if
at its Incidence its perpendicular Velocity was infinitely
little.

And the fame Propofition holds true of any Motion or
Thing perpendicularly retarded in its paflage through that
fpace, if inftead of the Summ of the two Squares you take
their difference. The Demonftration Mathematicians will
eafily find out, and therefore I fhall not trouble the Rea-
der with it.

Suppofe now that a Ray coming moll: obliquely in thepiv i.
Line MC be refrad:ed at C by the Plane RS into the Line
CN, and if it be required to find the Line CE into which
any other Ray AC fliall be refrac1:ed 3 let MC, AD, be
the Sines of incidence of the two Rays, and NG, EF, their
Sines of Refradtion, and let the equal Motions of the In-
cident Rays be reprefented by the equal Lines M C and
AC, and the Motion MC being confidered as parallel to
the refrading Plane, let the other Motion AC be diftin-
guifhed into two Motions AD and DC, one of which
AD is parallel, and the other DC perpendicular to the re-
fracting Surface. In like manner, let the Motions of the
emering Rays be diftinguifh'd into two, whereof the per-

H pendicular

C5BI

perpendicular ones are j^ CG and ^p CF. And if the

force of the refrading Plane begins to ad upon the Rays
either in that Plane or at a certain diftance from it on the
one fide, and ends at a certain diftance from it on the
other fide, and in all places between thofe two Limits ads
upon the Rays in Lines perpendicular to that rafi-ading
Plane, and the Adions upon the Rays at equal diftances
from the refrading Plane be equal, and at unequal ones ei-
ther equal or unequal according to any rate whatever 5
that motion of the Ray which is Parallel to the refrading
Plane will fuffer no alteration by that force ; and that mo-
tion which is perpendicular to it will be altered according
to the rule of the foregoing Propofition. If therefore for
the perpendicular Velocity of the emerging Ray CN you

write ^ CG as above, then the perpendicular Velocity
of any other emerging Ray CE which was ^ CF, will be

equal to the fquare Root of CD^ + -^^ CGq. And

by fquaring thefe equals, and adding to them the Equals
AD^ and MC^ — CD^, and dividing the Summs by the
Equals CVq -\- EVq and CG^ -|- NG^, you will have

Ypl equal to ^^. Whence AD, the Sine of Incidence,

is to EF the Sine of Refradion, as MC to NG, that is,
in a given ratio. And this Demonftration being general,
without determining what Light is, or by what kind of
force it is refraded, or afluming any thing further than
that the refrading Body ads upon the Rays in Lines per-
pendicular to its Surface y I take it to be a very convincing
Argument of the full Truth of this Propofition, ,

[59]

So tlien, if the ratio of the Sines of Incidence and Re-
fra<5tion of any fort of Rays be found in any one Cafe, 'tis
given in all Cafes 5 and this may be readily found by the
Method in the following Propoficion.

PROP. VII. Theor. VI.

Tlje TerfeBion of Tele/copes is impeded hy the dijferent ^frati-
gibility of the ^ys of Light.

TH E imperfedion of Telefcopes is vulgarly attri-
buted to the fpherical Figures of the Glaffes, and
therefore Mathematicians have propounded to Figure them
by the Conical Sedions. To fliew that they are mifta-
ken, I have inferted this Propofitionj the truth of which
will appear by the meafures of the Refraftions of the feve-
ral forts of Rays 5 and thefe meafures I thus determine.

In the third experiment of the firft Book, where the re-
frad:ing Angle of the Prifm was 6i\ degrees, the half of
that Angle 3 1 deg. 1 5 min. is the Angle of Incidence of
the Rays at their going out of the Glafs into the Air 3 and
the Sine of this Angle is 5188, the Radius being loooo.
When the Axis of this Prifm was parallel to the Horizon,
and the Refradion of the Rays at their Incidence on this
Prifm equal to that at their Emergence out of it, I obferved
with a Quadrant the Angle which the mean refrangible Rays
(that is, thofe which wentto the middle oftheSun s colour-
ed Image ) made with the Horizon and by this Angle and
the Sun's altitude obferved at the fame time, I found the
Angle which the emergent Rays contained with the incident
to be 44 deg. and 40 min. and the half of this Angle ad-
ded to the An^le of Incidence 3 i deg. 1 5 min. makes the

H 2 Angle

[do]

Angle of Refradion, which is therefore 5^ dcg. ^^ min. and
its Sine 8047. Thefe are the Sines of Incidence and Re-
fTa6tion of the mean refrangible Rays, and their proportion
in round numbers is 10 ta^ 1. This Glafswas of a colour in-
clining to green. The laft of the Prifms mentioned in the
third Experiment was of clear white Glafs. Its refrad:ing
Angle 63^ degrees. The Angle which the emergent Rays
contained, with the incident 45 deg. 50 min. The Sine of
half the firfl: Angle 5262. The Sine of half the Summ
of the Angles 8157. And their proportion in round num-
bers 20 to 31 as before.

From the Length of the Image, which was about 9I or

1 o Inches, fubdu6t its Breadth, which was 2 ^ Inches, and
the Remainder 7' Inches would be the length of the Image
were the Sun but a point, and therefore fubtends the An-
ale which the moft and leaft refrangible Rays, when inci-
dent on the Prifm in the fame Lines, do contain with one
another after their Emergence. Whence this Angle is

2 dcCT. 0/ 7." For the diftance between the Image and the
Prifm where this Angle is made, was 1 8 ~ Feet, and at that
diftance the Chord 7^ Inches fubtends an Angle of 2 deg.
o.' 7." Now half this Angle is the Angle which thefe e-
meraent Rays contain with the emergent mean refrangible
Rays, and a quarter thereof, that is 30. 2," may be ac-
counted the Angle which they would contain which the
fame emergent mean refrangible Rays, were they co-inci-
dent to them within the Glafs and fuffered no other Re-
fraction then that at their Emergence. For if two equal
Refracflions, the one at the incidence of the Rays on the
Prifm, the other at their Emergence, make half the Angle
1 deg. 0.' 7. then one of thofe Refradiions will make
about a quarter of that Angle, and this quarter added to

and

[6I]

and fubduded from the Angle of Refradion of the mean
refrangible Rays, which was 5 ^ deg. ^5', gives the An-
gles of Refra(5t:ion of the moft and leaft refrangible Rays
54 deg. 5' 2", and 53 deg. 4' 58", whofe Sines are 8099
and 7995, the common Angle of Incidence being 3 1 deg.
15' and its Sine 5188- and thefe Sines in the leaft round
numbers are in proportion to one another as 78 and 77
to 50.

Now if you fubdu(5l the common Sine of Incidence 50
from the Sines of Refra6lion 77 and 78, the remainders
17 and 28 (hew^ that in fmall Refractions the Refrad:ion
of the leaft refrangible Rays is to the Rcfra6tion of the moft
refrangible ones as 27 to 28 very nearly, and that the dif-
ference of the Refractions of the leaft refrangible and moft
refrangible Rays is about the 27^th part of the whole Re-
fraction of the mean refrangible Rays.

Whence they that are skilled in Opticks will eafily un-
derftand, that the breadth of the leaft circular fpace into
which Object' Glafles of Telefcopes can collect all forts of
Parallel Rays, is about the 27^th part of half the aperture
of the Glafs, or 55 th part of the whole aperture 3 and
that the Focus of the moft refrangible Rays is nearer to the
Object-Glafs than the Focus of the leaft refrangible ones, by
about the 27-^th part of the diftance between the Object-
Glafs and the Focus of the mean refrangible ones.

And if Rays of all forts,flowing from any one lucid point
in the Axis of any convex Lens, be made by the Refraction
of the Lens to converge to points not too remote from the
Lens , the Focus of the moft refrangible Rays fhall be
nearer to the Lens than the Focus of the leaft refrangible
ones, by a diftance which is to the 27^th part of the di-
ftance of the Focus of the mean refrangible Rays from the
Lens as the diftance between that Focus and the lucid

point

poiac from whence che Rays flow is to the diftance be-
tween that lucid point and the Lens very nearly.

Now to examine whether the difference between the Re-
fractions which the mofl: refrangible and the leaft refran-
gible Rays flowing from the fame point fufl^er in the Ob-
je(5t-GlaiTes ofTelefcopes and fuch like GlalTes, be fo great
. as is here defcribed, I contrived the following Experi-
; ment.

Exper. 1 6. The Lens which I ufed in the fecond and
-•eighth Experiments, being placed fix Feet and an Inch dif-
'tant from any Obje(ri;, colle(5led the Species of that Objed:
'by the mean refrangible Rays at the diflance of fix Feet
and an Inch from the Lens on the other fide. And there-
Tore by the foregoing Rule it ought to colle(fl: the Species of
that Object by the leafl refrangible Rays at the diflance of
'fix Feet and 3 - Inches from the Lens, and by the mofl re-
frangible ones at the diflance of five Feet and lo^ Inches
from it : So that between the tvi o Places where thefe leafl
and mofl refrangible Rays colle6t the Species, there may
be the diflance of about 5-j Inches. For by that Rule, as
>fix Feet and an Inch ( the diflance of the Lens from the
lucid Object ) is to twelve Feet and two Inches ( the di-
■ftance of the lucid Object from the Focus of the mean re-
frangible Rays) that is, as one is to two, fo is the 27 ^th
part of fix Feet and an Inch (the diflance between the Lens
and the fame Focus ) to the diflance between the Focus of
'the mofl refrangible Rays and the Focus of the leafl re-
frangible ones, which is therefore 5 - Inches, that is very
-nearly 5 '- Inches. Now to know whether this meafure
was true, I repeated the fecond and eighth Experiment of
"this Book with coloured Light, which was lefs compound-
..ed than that I there made ufe of : For I now feparated the

hetero-

h eterogeneous Rays from one another by the Method I de-
fcribed in the i ith Experiment, fo as to make a coloured
Spedrum about twelve or fifteen times longer than broad.
This Spedlrum I caft on a printed book, and placing the
above-mentioned Lens at the diftance of fix Feet and an
Inch from this Spectrum to colled: the Species of the illu-
minated Letters at the fame difiiance on the other fide, I
found that the Species of the Letters illuminated with Blue
were nearer to the Lens than thofe illuminated with deep
Red by about three Inches or three and a quarter : but the
Species of the Letters illuminated with Indigo and Violet
appeared fo confufed and indiftind, that I could not read
them : Whereupon viewing the Prifm, I found it was full
of Veins running from one end of the Glafs to the other j
fo that the Refradion could not be regular. I took ano-
ther Prifm therefore which was free from Veins, and in-
ftead of the Letters I ufed two or three Parallel black Lines
a little broader than the ftroakes of the Letters, and caft-
ing the Colours upon thefe Lines in fuch manner that the
Lines ran along the Colours from one end of the Spedium
to the other, I found that the Focus where the Indigo, or
confine of this colour and Violet call the Species of the
black Lines moft difi:!nd:ly,tobe about 4 Inches or 4^ near-
er to the Lens than the Focus where the deepeft Red cafi;
the Species of the fame black Lines mofl: diftindly.
The violet was fo faint and dark, that I could not
difcern the Species of the Lines diftinctly by that Co-
lour J and therefore confidering that the Prifm was made
of a dark coloured Glafs inclining to Green, I took another
Pifm of clear white Glafs • but the Spedrum of Colours
which this Prifm made had long white Streams of faint
Light fliooting out from both ends of the Colours, which
made me conclude that fomefhing was amifs j and view-
ing

[64]

ing the Prifm, I found two or three little Bubbles in the
Glafs which refracted the Light irregularly. Wherefore I
covered that part of the Glafs with black Paper, and let-
ting the Light pafs through another part of it which was
free from fuch Bubles, the Sped:rum of Colours became
free from thofe irregular Streams of Light, and was now
fuch as I defired. But ilill I found the Violet fo dark and
faint, that I could fcarce fee the Species of the Lines by the
Violet, and not at all by the deepeft part of it, which was
next the end of the Spectrum. I fulpedled therefore that
this faint and dark Colour might be allayed by that fcat-
tering Light which was refracted, and reflected irregularly
partly by fome very fmall Bubbles in the Glafles and
partly by the inequalities of their Polifli: which Light,
tho' it was but little, yet it being of a White Colour,
might fuffice to affect the Senfe fo ftrongly as to difturb
the Pha^nomena of that weak and dark Colour the Violet,
and therefore I tried, as in the 12th, i^th, 14th Experi-
ments, whether the Light of this Colour did not confift of
a fenfible mixture of heterogeneous Rays, but found it did
not. Nor did the Refractions caufe any other fenfible
Colour than Violet to emerge out of this Light, as they
would have done out of White Light, and by con-
fequence out of this Violet Light had it been fenfi-
bly compounded with White Light. And therefore Icon-
eluded, that the reafon why I could not fee the Species of
the Lines diflindtly by this Colour, was only the darknefs
of this Colour and Thinnels of its Light, and its dif-
tance from the Axis of the Lens 3 I divided therefore thofe
Parallel Black Lines into equal Parts, by which I might
readily know the diflances of the Colours in the Spedirum
from one another, and noted the diftances of the Lens
from the Foci of fuch Colours as cafl the Species of the

Lines

Lines diftindly, and then confidered whether the diffe-
rence of thofe diftances bear fuch proportion to 5 '^Inches,
the greateft difference of the diftances which the Foci of
the deepeft Red and Violet ought to have from the Lens,
as the diftance of the obferved Colours from one another
in the Spedrum bear to the like diftance of the deepeft Red
and Violet meafured in the redlilinear fides of the Spect-
rum, that is, to the length of thofe fides or excefs of the
length of the Spectrum above its breadth. And my Ob-
fervations were as follows.

When I obferved and compared the deepeft fenfibleRed,
and the Colour in the confine of Green and Blue, which
at that rectilinear fides of the Spe(ftrum was diftant from it
half the length of thofe fides, the Focus where the confine
of Green and Blue caft the Species of the Lines diftin(5tly
on the Paper, was nearer to the Lens then the Focus where
the Red caft thofe Lines di(5tin6tly on it by about i^ or
2 .' Inches. For fometimes the Meafures were a little grea-
ter, fomctimes a little lefs, but feldom varied from one
another above j of an Inch. For it was very difficult to
define the Places of the Foci, without fome little Errors.
Now if the Colours diftant half the length of the Image,
( meafured at its rectilinear fides ) give 2^; or 2 - difference
of the diftances of their Foci from the Lens, then the Co-
lours diftant the whole length ought to give 5 or 5I Inches
difference of thofe diftances.

But here it's to be noted, that I could not fee the Red
to the full End of the SpeCtrum, but only to the Center
of the Semicircle which bounded that End, or a little far-
ther 3 and therefore I compared this Red not with that Co-
lour which was exactly in the middle of the SpeCtrum, or
confine of Green and Blue, but with that which veracd a
little more to the Blue than to the Green : And as I reck-

I oned

[66]

oried the whole length of the Colours not tob^ the whole
length of the Spe(artim, but the leflgth of its reftiHnear
fides, fo completing theSemicirlar Ends into Circles, when
(iither of the obferved Colours fell within thofe Circles, I
meafured the diftance of that Colour from the End of the
Spedrum, and fubdu(fiing half the diftance from the mea-
fured diftance of the Colours, I took the remainder for
their cbrre(5ted diftance 3 and in thefe Obfervations fet
down this correded diftance for the difference of their di-
ftances from the Lens. For as the length of the redilinear
fides of the Spectrum would be the whole length of all the
^Colours, were the Circles of which ( as we fhewed) that
Spedrum confifts contra(5led and reduced to Phyfical
Points, fo in that Cafe this correded diftance would be the
real diftance of the obferved Colours.

When therefore I further obferved the deepeftfenfible Red,
and that Blue whofe corre<5led diftance from it was ^ parts
of the length of the redilinear fides of the Spectrum, the
difference of the diftances of their Foci from the Lens was
about ^- hiches, and as 7 to i 2 fo is 3 -J to 5 i.

When I obferved the deepeft fenfible Red, and that Indi-
go whofe corrected diftance was ^ or J of the length of the
rectilinear fides of the Spe6trum, the difference of the di-
ftances of their Foci from the Lens, was about 3 '* Inches,
and as 2 to 3 fo is 3 J-to '){.

When I obferved the deepeft fenfible Red, and that deep
Indigo whofe corrected diftance from one another was ^ or
■' of the length of the redilinear fides of the Spedum, the
difference of the diftances of their Foci from the Lens was
about 4 Inches 3 and as 3 to 4 fo is 4 to 5 J.

When I obferved the deepeft fenfible Red, and that part
of the Violet next the Indigo whofe correded diftance from
the Red was {^ or j of the length of the redilinear fides of

the.

the Spedtrum, the difference of the diftances <)f their Foci
from the Lens was about 4^ Inches j and as 5 to ($, fo is
4- to 5-. For fometimes when the Lens was advantagi-
oufly placed, fo that its Axis relpeded the Blue, and all
things elfe were well ordered, and the Sun fhone clear, and
I held my Eye very near to the Paper on which the Lens
caft the Species of the Lines, I could fee pretty diftinctly
the Species of thofe Lines by that part of the Violet which
was next the Indigo ; and fometimes I could fee them by
above half the Violet. For in making thefe Experiments
I had obferved, that the Species of thofe Colours only ap-
peared dijftinct which were in or near the Axis of the Lens :
So that if the Blue or Indigo were in the Axis, I could fe,e
their Species diilinctly ; and then the Red appeared much
lefs diftinct than before. Wherefore I contrived to majce
the Spectrum of Colours fliorter than before, fo that both
its Ends might be nearer to the Axis of the Lens. And
now its length was about 2^ Inches and breadth about -or
I of an Inch. Alfo inftead of the black Lines on which the
Spectrum was caft, I made one black Line broader than
thofe, that I might fee its Species more eafily ; and this
Line I divided by fhort crofs Lines into equal Parts, for
tneafuring the diftances of the obfervedColours. And now
I could fometimes fee the Species of this Line v/ith its divi-
iions almoft as far as the Centers of the Semicircular Violet
End of the Spectrum, and made thefe further Qbfervations.
When I oblerved the decpeft fenfible Red, and that part
of the Violet whofe corrected diftance from it was about
j Parts of the rectilinear fides of the Spe6lrum the difference
•of the diftances of the Foci of thofe Colours from the Lens,
was one time 4-*, another time 4^, anothertimc 4^, Inches,
andasS to 9, foare4j, 4-;, 4I, to 5', ^^^5^^ refpedively.

I 2 When

[68]

When I obferved tlie deepeft fenfible Red, and deepefi:
fenfible Violet, (the corrected dlftance of which Colours-
when all things were ordered to the bed advantage, and the
Sun fhone very clear, was about ^ or ^ parts of the length
of the rectilinear fides of the coloured Spectrum, ) I found
the difference of the diftances of their Foci from the Lens
fometimes 4.' fometimes 5-, and for the mofl part 5 Inches
or thereabouts : and as 11 to i 2 or 15 to i6, fo is five
Inches to 5 ■; or 5 i Inches.

And by this progreflion of Experiments I fitisfied my
felf, that had the light at the very Ends of the Spectrum been
ftrong enough to make the Species of the black Lines ap-
pear plainly on the Paper, the Focus of the deepeft Vic-
let would have been found nearer to the Lens, than the Fo-
cus of the deepeft Red, by about y- Inches at leaft. And
this is a further Evidence, that the Sines of Incidence and
Refra6tion of the feveral forts of Rays, hold the fame pro-
portion to one another in the fmalleft Refracftions which
they do in the greateft.

My progrefs in making this nice and troublefome Expe-
riment I have fet down more at large, that they that fliall
try it after me may be aware of the Circumfpedtion re-
quifite to make it fucceed well. And if they cannot make
It fucceed fo well as I did, they may notwithftanding col-
led: by the Proportion of the diftance of the Colours in the
Spedrum, to the difference of the diftances of their Foci
from the Lens, what would be the fuccefs in the more di-
ftant Colours by a better Trial. And yet if they ufe a
broader Lens than I did, and fix it to a long ftreight Staff
by means of which it may be readily and truly direded to
the Colour whofe Focus is defired, I queftion not but the
Experiment will fucceed better with them than it did with
me,. For I direded the Axis as nearly as I could to the

middle-:

middle of the Colours, and then the faint Ends of the
Spedrum being remote from the Axis, caft their Species lefs
diftindly on the Paper than they would have done had the
Axis been fucce/fively dire(5i:ed to them.

Now by what has been faid its certain, that the Rays
which differ in refrangibility do not converge to the fame
Focus, but if they flow from a lucid point, as far from
the Lens on one fide as their Foci are one the other, the
Focus of the moft refrangible Rays fliall be nearer to the
Lens than that of che leafl refrangible, by above the four-
teenth part of the whole diftance: and if they flow from a lu-
cid point, fo very remote from the Lens that before their
Incidence they may be accounted Parallel, the Focus of the
mofl: refrangible Rays fliall be nearer to the Lens than the
Focus of the leafl: refrangible, by about the 27th onSth part
of their whole difliance from it. And the Diameter of the
Circle in the middle fpace between thofe two Foci which
they illuminate when they fall there on any Plane, perpen-
dicular to the Axis (which Circle is the leafl; into which
they can all be gathered) is about the 55th part of the Dia-
meter of the aperture of the Glafs. So that 'tis a wonder
that Telefcopes reprefent Objeds fo difl:in<5i: as they do. But
were all the Rays of Light equally refrangible, the Error
arifing only from the fphericalnefs of the Figures of Glafles
would be many hundred times lefs. For if the Objed:-
Glafsofa Telefcope be Plano-convex, and the Plane fide
be turned towards the Objed, and the Diameter of the
Sphere whereof this Glafs is a fegment,be called D, and the
Semidiameter of the aperture of the Glafs be called S, and
the Sine of Incidence out of Glafs into Air, be to the Sine of
Refradlion as I to R : the Rays which come Parallel to the
Axis of the Glafs, fliall in the Place where the Image of the
Objed is mofl: diftindtly made, be fcattered all over a little

Circle

[70]

Circle whofe Diameter is ^ '^ Dfi^d. ^^^f ^^^^^Yj ^^ ^ ga-
ther by computing the Errors of the Rays by the method
of infinite Series, and rejeding the Terms whofe cjuanti-
tities are inconfiderable. As for inftance, if the Sine of In-
cidence I, be to the Sine of Refradion R, as 20 to ; i, and
if D the Diameter of the Sphere to which the Convex fide
of the Glafs is ground, be 100 Feet or 1200 Inches, and
S the Semidiameter of the aperture be two Inches, the

Diameter of the little Circle ( that is fi^^j^ ) will be

— ^^— ( or ,z-^l„^^ ) parts of an Inch. But the

20 X 1200 x 1200 *■ 5600000 •' r

Diameter of the little Circle through which thefe Rays are
Scattered by unequal refrangibility, will be about the 55 th
part of the aperture of the Objed:-Glafs which here is four
Inches. And therefore the Error arifing from the fpherical
Figure of the Glafs, is to the Error arifing from the diffe-
rent Refrangibility of the Rays, as ^^^^ to ^ that is as i
to 8151 : and therefore being in Comparifon fo very little,
deferves not to be confidered.

But you will fay, if the Errors caufed by the different re-
frangibility be fo very great, how comes it to pafs that Ob-
jed;s appear through Telefcopcs fo diilinfl as they do ? I an-
fwer, 'tis becaufe the erring Rays are not fcattered uniform-
ly over all that circular fpace, but collected infinitely more
denfely in the Center than in any other part of the Circle,
and in the way from the Center to the Circumference grow
continually rarer and rarer, fo as at the Circumference to
become infinitely rare 3 and by reafon of their rarity are
p. not ftrong enough to bevifible, unlefs in the Center and ve-

^' ry near it. Let ADE reprefent one of thofe Circles de-
fcribed with the Center C and Semidiameter AC, and let
BFG be afmaller Circle concentric to the former, cutting

with

[71]

with its Circumference the Diameter AC in B, and befecc
AC in N, and by my reckoning the denfity of the Light
in anyplace B will be to its denfity inN, as AB to BC,
and the whole Light within the leffer Circle BFG, will be
to the whole Light within the greater AED, as the Excefs of
the Square of AC above the Square of AB, is to the Square
of AC. As if BC be the fifth part of AC, the Light will be
four times denfer in Bthan in N, and the whole Light with-
in the lefs Circle,will be to the whole Light within the grea-
ter, as nine to twenty five. Whence it's evident that the
Light within the lefs Circle, muflflrike the fenfe much more
ftrongly, than that faint and dilated light round about be-
tween it and the Circumference of the greater.

But its further to be noted, that the mofl luminous of
the prifmatick Colours are the Yellow and Orange. Thefe
ailed: the Senfes more ftrongly than all the refl together, and
next to thefe in ftrength are the Red and Green. The Blue
compared with thefe is a fiinc and dark Colour, and the In«f
digo and Violet are much darker and fainter, fo that thefe
compared with the ftronger Colours are little to be regard-
ed. The Images of Objedis are therefore to be placed, not
in the Focus of the mean refrangible Rays which are in the
confine of Green and Blue, but in the Focus of thofe Rays
which are in the middle of the Orange and Yellow 3 there
where the Colour is moft luminous and fulgent, that is in
the brighteft Yellow, that Yellow which inclines more to
Orange than to Green. And by the Refradion of thefe
Rays ( whofe Sines of Incidence and Refradion in Glafs
are as 1 7 and 11) the Refradion of Glafs and Cryftal for
optical ufes is to be meafured. Let us therefore place the
Image of the Objed in the Focus of thefe Rays, and all the
Yellow and Orange will fall within a Circle, whofe Dia-
ineter is about the 250th part of the Diameter of the aper-
ture

[72]

ture of the Glafs. And if you add the brighter half of the
Red, ( that half which is next the Orange, and the brighter
half of the Green, (that half which is next the Yellow,) a-
bout three fifth parts of the Ijght of thefe two Colours will
fall within the fame Circle,and two fifth parts will fall with-
out it round about ; and that which falls without will be
Ipread through almoft as much more fpace as that which
falls within, and fo in the grofs be almoft three times ra-
rer. Of the other half of the Red and Green, ( that is of
the deep dark Red and Willow Green ) about one quarter
will fall within this Circle, and three quarters without, and
that which falls without will be fpread through about four
or five times more fpace than that which fall within; and fo
in the grofs be rarer, and if compared with the whole Light
within it, willbe about 25 times rarer than all that taken in
the grofs ; or rather more than 30 or 40 times rarer, be-
caufe the deep red in the end of the Spedrum of Colours
made by a Prifm is very thin and rare, and the Willow Green
is fomething rarer than the Orange and Yellow. The Ltght
of thefe Colours therefore bring fo very much rarer than that
within the Circle, will fcarce afFed: the Senfe efpecially fince
the deep Red and Willow Green of this Light, are much
darker Colours then the reft. And for the lame reafon the
Blue and Violet being much darker Colours than thefe, and
much more rarified, may be neglected. For tlie denfe and
bright Light of the Circle, will obfcure the rare and weak
Light of thefe dark Colours round about it, and render them
almoft infenfible. The fenfible Image of a lucid point is
therefore fcarce broader than a Circle whofe Diameter is
the 250th part of the diameter of the aperture of the Object
Glafs of a good Telefcope, or not much broader, if you
except a faint and dark mifty light round about it, which
a Spectator will fcarce regard. And therefore in a Telefcope

whofe

[7?] . ,

vvhofe aperture is four Inches, and length an hundred Feet/
it exceeds not 2 '45', or 5". And in a Telefcope whofc
aperture is two Inches, and length 20 or 30 Feet, it may
be 5 "or 6" and fcarce above. And this Anfwers well to
Experience : For fome Aftronomers have found the Dia-
meters of the fixt Stars, in Telefcopes of between twenty
and fixty Feet in length, to be about 4' or 5" or at moft
6" in Diameter. But if the Eye-Glafs be tinded faintly
with the fmoke of a Lamp or Torch, to obfcure the Light
of the Star, the fainter Light in the circumference of the
Star ceafes to be vifible, and the Star (if the Giafs be fuffici-
ently foiled with fmoke) appears fomething more like a Ma-
thematical Point. And for the fame reafon, the enormous
part of the Light in the Circumference of every lucid Point
ought to be lefs difcernable in fhorter Telefcopes than in
longer, becaufe the fhorter tranfmit lefs Light to the Eye.

Now if we fuppofe the fenlible Image of a lucid point,
to be even 250 times narrower than the aperture of the
Glafs: yet were it not for the different refrangibility of the
Rays, its breadth in an 1 00 Foot Telefcope whofe aperture
is 4 Inches would be but ^-^^^^ parts of an Inch, as is ma-
nifeft by the foregoing Computation. And therefore in
this Cafe the greateft Errors arifing from the fpherical Figure
of the Glafs, would be to the greateft fenfible Errors ari-
fing from the different refrangibility of the Rays as -53^
to ^-^ at moft, that is only as i to 1826. And this fuffi-
ciently fliews that it is not the fpherical Figures of Glaffes
but the different refrangibility of the Rays which hinders the
perfection of Telefcopes.

There is another Argument by which it may appear that
the different refrangibility of Rays, is the true Caufe of the
imperfedion of Telefcopes. For the Errors of the Rays
arihng from the fpherical Figures of Objed-Glafles, are as

K the

[ 74 3

tlic Cubes of the apeitures of the Objed^Glaflesjand thence
to make Telefcopes of various lengths, magnify with equal
cUftindnefs, the apertures of the Objed-GlafTes, and the
C)harges or magnifying Powers, ought to be as the Cubes of
the fcjuare Pvoots of their lengths 5 which doth not anfwer
10 Experience. But the errors of the Rays arifing from
the d liferent refrangibility, are as the apertures of the Ob-
jetft-Glafies, and thence to make Telefcopes of various
leno^ths, magnify' with equal diftin(5lnefs, their apertures and
charges ought to be as the fquare Roots of their lengths -,.
and this aniwers to experience as is well known. For in-
ftance, a Telefcope of 64 Feet in length, with an aperture
of 1- Inches, magnifies about i 20 times, with as much dif-
tind:nefs as one of a Foot in length, with j of an Jnch aper*
ture, magnifies i 5 times.

Now were it not for this different refrangibility of Rays,
Telefcopes might be brought to a greater Perfedion than
we have yet defcribed, by compoffng the Objed-Glafs of
two Glafles with Water between them. Let ADFC repre^
f^. 2 8.|-ent the Objed-Glafs compofed of two Glaffes ABED and
and BEFC, alike convex on the outfides AGD and CHF,
and alike concave on the infides BME, BNE, with Water
in the concavity BMEN. Let the Sine of Incidence out of
Glafs into Air be as I to R and out of Water into Air as K
to R, and by confequence out of Glafs into Water, as I to
K : and let the Diameter of the Sphere to which the convex
fides AGD and CHF are ground be D, and the Diameter,
of the Sphere to which the concave fides BME and BNE
are ground be to D, as the Cube Root of KK— KI to the
Cube Root of RK— RI: and the Refra^ions on the con-
cave fides of the Glafles, will very much corrcd: the Errors
of the Refractions on the convex fides, fo far as they arife
from the fphericainefs of the Figure. And by this means

might

[75]

might Telefcopes be brought to fufficient perfe6bion, wercit
not' for the diflferentrefrangibility of feveralforsof Rays. But
by reafon of this different refrangibility, I do not yet fee any
other means of improving Telefcopes by Refradlions alone
than that of increafing their lengths, for which end the late
contrivance of Hugenius feems well accommodated. For
very long Tubes are cumberfome, and fcarce to be readily
managed, and by reafon of their length are very apt to
bend, and fhake by bending fo as to caufe a continual
trembling in the Objects, whereby it becomes difficult to
fee them diftindly : whereas by his contrivance the Glaffes
are readily manageable, and the Objedl-Glafs being fixt up-
on a ftrong upright Pole becomes more fteddy.

Seeing therefore the improvement of Telefcopes of given
lengths by Refractions is defperate 3 I contrived heretofore a
Perfpedlive by reflexion, ufing inftead of an Objed: Glafs
a concave Metal. The diameter of the Sphere to which
the Metal was ground concave was about 2 5 Englifli Inches,
and by confe^uence the length of the Inftrument about (i3t
Inches and a quarter. The Eye-Glafs was plano-convex,
and the Diameterof the Sphere to which the convex fide was
ground was about i of an Inch, or a little lefs, and by con-
fequence it magnified between ^ o and 40 times. By ano-
ther way of meafuring I found that it magnified about
^ 5 times. The Concave Metal bore an aperture of an Inch
and a third part j but the aperture was limited not by an
opake Circle, covering the Limb of the Metal round about,
but by an opake circle placed between the Eye-Glafs and the
Eye, and perforated in the middle with a little round hole
for the Rays to pafs through to the Eye. For this Circle
by being placed here, ftopr much of the erroneous Light,
which otherwife would have difturbed the Vifion. By com-
paring it with a pretty good Perfpedive of four Feet in

K 2 length,

length, made with a concave Eye-Glafs, I could read at x
greater diftance with my own Inftrument than with the
Glafs. Yet Objedts appeared much darker in it than in the
Glafs, and that partly becaufe more Light was loft by re-
flexion in the Metal, then by refrailion in the Glafs, and.
partly becaufe my Inftrument was overcharged. Had it
magnified but ^oor 25 times it would have made the Object
appear more brisk and pleafant. Two of thefelmade about:
16 Years ago, and have one of them ftill by me by which
"li can prove the truth of what I write. Yet it is not fo good
as at thefirft> For the concave has been divers times tar-
niflied and cleared again, by rubbing it with very foft Lea^-
ther. When I made thefe, an Artift in London undertook;
to imitate it 5 but ufing another way of polifliing them
than I did, he fell much fhort of what I had attained to,.
as I afterwards underftood by difcourfing the under- Work-
man he had imployed. The Polifli I ufed was on this man-
ner. I had two round Copper Plates each fix Inches in:
Diameter, the one convex the other concave, ground ve-
ry true to one another. On the convex I ground the Ob-
jedl-Metal or concave which was to be polifh'd, till it had.
taken the Figure of the convex and was ready for a Polifh.
Then I pitched over the convex very thinly, by dropping
melted pitch upon it and warming it to keep the pitch
foft, whilft I ground it with the concave Copper wetted to
make it fpread evenly all over the convex. Thus by work-
ing it well I made it as thin as a Groat, and after the con-
vex was cold I ground it again to give it as true a Figure as
I could. Then I took Putty which I had made very fine
by wafliing it from all its grofler Particles, and laying a lit-
tle of this upon the pitch, I ground it upon the Pitch with
the concave Copper till it had done making a noife j and
then upon the Pitch I ground the Objed;-Mecal with a brisk

Motion

[77]

Motion, for about two or three Minutes of time, leaning
hard upon it. Then I put frefh Putty upon the Pitch and
ground it again till it had done making a noife, and after-
wards ground the Obje<ft Metal upon it as before. And
this Work I repeated till the Metal was polifhed, grinding
it the lad time with all my flrength for a good while toge-
the/, and frequently breathing upon the Pitch to keep ir
moift without laying on any more frefh Putty. The Ob-
ject-Metal was two Inches broad and about one third part
of an Inch thick, to keep it from bending. I had two of
thefe Metals, and when I had polifhed them both I tried
which was beft, and ground the other again to fee if I could
make it better than that which I kept. And thus by many
Trials I learnt the way of poliiliing, till I made thofe two
refledling Peipe6lives I fpake of above. For this Art of
polifliing will be better learnt by repeated Practice than by
my defcription. Before I ground the Objcvft Metal on the
Pitch, r always ground the Putty on it with the concave
Copper till it had done making a noife, becaufe if the Par-
ticles of the Putty were not by this means made to flick
fafl in the Pitch, they would by rolling up and down grate
and fret the Objed Metal and fill it full of little holes.

But becaufe Metal is more difficult to polifh than Glafs
and is afterwards very apt to be fpoiled by tarnifliing, and
refle(5ts not fo much Light as Glafs quick-filvered over does:
I would propound touleinfleadof theMetal, a Glafs ground
concave on the forefide, and as much convex on the back-
fide, and quick-filvered over on the convex fide. The Glafs
mufl be every where of the fame thicknefs exactly. Other-
wife it will make Objedls look coloured and indiflind. By
fuch a Glafs I tried about five or fix Years ago to make
a refliediing Telefcope of four Feet in length to magnify a-
bout 1 50 times, and I fatisfied my felf that there wants no-
thing

C78]

thincT but a good Artift to bring the defign to Perfe(flion.
For the Glafs being wrought by one of our London Artifts
after fuch a manner as they grind Glafles for Telefcopes,
tho it feemed as well wrought as the Objed: Glafles ufe to
be, yet when it was quick-nlvered, the reflexion difcovered
innumerable Inequalities all over the Glafs, And by reafon
of thefe Inequalities, Objeds appeared indifliindl in this In-
flrument. For the Errors of refletfted Rays caufed by any
Inequality of the Glafs, are about fix times greater than the
Errors of refraded Rays caufed by the like Inequalities. Yet
by this Experiment I fatisfied my felf that the reflexion on
the concave fide of the Glafs, which I feared would difturb
the vifion,didno fenfible prejudice to it, and by confequencc
that nothing is wanting to perfed thefe Telefcopes, but
good Workmen who can grind and polifti Glafles truly fphe-
rical. An Objedi-Glafs of a fourteen Foot Telefcope, made
by one of our London Artificers, I once mended confidera-
bly, by grinding it on Pitch with Putty, and leaning ve-
ry eafily on it in the grinding, lefl: the Putty fliould fcratch
it. Whether this way may not do well enough for poliflv-
ing thefe reflecting Glafles, I have not yet tried. But he
that fhall try either this or any other way of polifhing which
he may think better, may do well to make his Glafles rea-
dy for polifliing by grinding them without that violence,
wherewith our London Workmen prefs their Glafles in grind-
ing. For by fuch violent preflure, Glafles are apt to bend
a little in the grinding, and fuch bending will certainly fpoil
rheir Figure. To recommend therefore the confideration
of thefe refleding Glafles, to fuch Artiflis as are curious in
figuring Glafles, I fhall defcnbe this Optical Infl:rument in
the following Propofition.

PROP.

l79l
PROP. VII. Prob. II.

To Jhorlen Tetef copes.
ET ABDC reprefcnt a Glafs fphcrically concave on p;^ ^^^

_j the forefide AB, and as much convex on the back-
fide CD, fo that it be every where of an equal thicknefs. Let
k not be thicker on one fide than on the other, left it make
Objei5ts appear coloured and indiftincH:, and let it be very
truly wrought and quick-filveredoveron the backfide 3 and
fet in the Tube VXYZ which mull be very black within.
Let EFG reprefent a Prifm of Glafs or Cryllal placed near
the other end of the Tube, in th^ middle of it, by means of
a handle of Brafs or Iron FGK, to the end of which made
flat it is cemented. Let this Prifm be rectangular at E, an-d
let the other two Angles at F and G be accurately equal to
each other, and by confeciuence equal to half right ones, and
let the plane fides FE and GE be fquare, and by confe-
queiuce the third fideFG a rectangular parallelogram, whofe
lenCTch is to its breath in a fubduplicate proportion of two
to one. Let it be fo placed in the Tube, that the Axis of
the Speculum may pats through the middle of the fquare
fide EF perpendicularly, and by confequence throuoh the
middle of the fide F G at an Angle of 45 degrees, and let the
fide EF be turned towards the Speculum, and the diftance
ofthis Prifm from the Speculum be fuch that the Rays of the
light PQ., RS, 8lc, which are incident upon theSpecuhim in
Lines Parallel to the Axis thereof, may enter the Prifm at
the fide EF, and be refleiCled by the fide F G, and thence
go out of it through the fide GE, to the point T which
muft be the common Focus of the Speculum ABDC, and of
a Plano-convex Eye- Glafs H, through which thole Rays
muft pafs to the Eye. And let the R.ays at their cominjr

out

[8o]

out of the Glafs pafs through a fmall round hole, or aper-
ture made in a little Plate of Lead, Rrafs, or Silver, where-
with the Glafs is to be covered, which hole muft be no
bigger than is necelfary for light enough to pafs through.
For fo it will render the Obje6t diftind, the Plate in which
'tis made intercepting all the erroneous part of the Light
v\ hich comes from the Verges of the Speculum AB. Such
an Inftrument well made if it be 6 Foot long, ( reckoning
the length from the Speculum to the Prifm, and thence to
the Focus T) will bear an aperture of 6 Inches at the Spe-
culum, and magnify between two and three hundred times.
But the hole H here lim.its the aperture with more advan-
tage, then if the aperture was placed at the Speculum. If
the Inftrument be made longer or fhorter, the aperture muft
be in proportion as the Cube of the fquare Root of the
length, and the magnifying as the aperture. But its con-
venient that the Speculum be an Inch or two broader than
the aperture at the leaft, and that the Glafs of the Speculum
be thick, that it bend not in the working. The Prifm EFG
muft be no bigger than is neceffary, and its back fide FG
muft not be guick-filvered over. For without quick-filver
it will refle6t all the Light incident on it from the Speculum.
In this Inftrument the Objed: will be inverted, but may
be ereded by making the fquare fides EF and EG of the
Prifm EFG not plane but fpherically convex, that the Rays
may crofs as well before they come at it as afterwards be-
tween it and the Eye-Glafs. If it be defired that the Inftru-
ment bear a larger aperture, that may be alfo done by com-
pofing the Speculum of two Glafles with Water between
them.

THE

Bookl.Pklrl.Ri

Fi&3.

Fig. 6.

T E

t q

^^g-7.

Fie: 8.

BookI.PlateJr.RrtI.

F'^S-9-

..<f-f-

•

Bookl.Plate in. Parti

BoOK,l.Plate,IV. Pa,

•

G H I K I. M

a- ^ c il c y

p a.::::::n:z:::Jj:zzjrz:::n::::::x) t
e A I k I m

Fig. 26.

t T-

2yP iP T

■,T7?1

/^/t? 19-

DEF

J...

6\^

^■f

T H E

FIRST BOOK

O F

O P T I C K S.

PART II.

PROP. I. THEOR. I.

T'he 'ph^enomena of Colours in refraSed or rejleBed Light
are not caujed b>j new modifications of the Light variouf'
ly imfref^ according to the variom terminations of the
Light and Shadoisj.

The Proof hy Experiments.

EX PER. I.

FOR if the Sua fhine into a very dark Chamber _pio-. i,
through an oblong Hole F, whofe breadth is the
iixth or eighth part of an Inch, or fomething lets ; and
his Beam FH do afterwards pafs hrft through a very
large Prifm ABC, diftant about 20 Feet from the

L Hole,

Hole, and parallel to it, and then (with its white part)
through an oblong Hole H, whole breadth is about
the fortieth or lixtieth part of an Inch, and which is
made in a black opake Body G I, and placed at the
diftance of two or three Feet from the Prifm, in a pa-
rallel fituation both to the Prifm and to the former
Hole, and if this w^hite Light thus tranfmitted through
the Hole H, fall afterwards upon a white Paper pt,.
placed after that Hole H, at the diftance of three or
four Feet from it, and there paint the ufual Colours of
the Prifm, fuppofe red at t, yellow at s, green at r,
blue at q, and violet at p ; you may with an iron Wire,
or any fuch like flender opake Body, whole breadth is
about the tenth part of an Inch, by intercepting the rays
at k, 1, m, noro, take away any one of the Colours
at t, s, r, q or p, whilft the other Colours remain up-
on the Paper as before ; or with an obftacle fomething
bigger you may take away any two, or three, or four Co-
lours together, the reft remaining: So that anyone of
the Colours as well as violet may become outmoft in
the confine of the fhadow towards p, and any one of
them as well as red may become outmoft in the confine
of the ftiadow towards t, and any one of them may alfo
border upon the ftiadow made within the Colours by
the obftacle R intercepting fome intermedia^te part of
the Light ; and, laftly, any one of them_ by-* being
left alone may border upon the ftiadow on either hand.
All the Colours have themfelves indifferently to any
confines of ftiadow, and therefore the differences of thefe
Colours from one another, do not arife from the diffe-
rent confines of ftiadow, whereby Light is varioufty
modified as has hitherto been the Opinion of Philofo-

phers.

[83]

pliers. In trying thefe things 'tis to be obferved, that
by how much the Holes F and H are narrower, and the
intervals between them, and the Prifm greater, and the
Chamber darker, by lb much the better doth the Ex-
periment iucceed ; provided the Light be not fo far
diminifhed, but that the Colours at pt be fufficiently
vilible. To procure a Pritm of folid Glafs large enough
for this Experiment will be difficult, and therefore a
prifmatick VelTel muft be made of polifhed Glafs-plates
cemented together, and hlled with Water.

EX PER. 11.

The Sun's Light let Into -a dark Chamber through Fig'. 2.
the round Hole F, half an Inch wide, pafled firft through
tlie Prifm ABC placed at the Hole, and then through
a Lens PT fomething more than four Liches broad, and
about eight Feet diftant from the Prifm,and thence con-
verged to O the Focus of the Lens diftant from it about
three Feet, and there fell upon a white Paper D E. If
that Paper was perpendicular to that Light incident up-
on it, as 'tis repreiented in the pofture D E, all the Co-
lours upon it at O appeared white. But if the Paper
being turned about an Axis parallel to the Prifm, be-
came very much inclined to the Light as 'tis reprefen-
ted in the politions de and o\ j the fame Light in the
one cafe appeared yellow and red, in the other blue.
Here one and the fame part of the Light in one and the
fame place, according to the various inclinations of the
Paper, appeared in one cafe white, in another yellow
or red, in a third blue, w^hilft the confine of Light and

L 2 Shadow,

[84-]

Shadow, and the refradions of the Prifm in all thefe
cafes remained, the fame.

EXPER. III.

Fisr. ^. Such another Experiment may be more ealily tried
as follows. Let a broad beam of the Sun's Light coming
into a dark Chamber through a Hole in the Window
fhut be refraded by a large Prifm ABC, whofe re-
frading Angle C is more than 60 degrees, and fo foon
as it comes out of the Prifm let it fail upon the white
Paper D E glewed upon a ftiif plane, and this Lighty
when the Paper is perpendicular to it, as 'tis reprelen-
ted in DE, will appear perfectly white upon the Paper,,
but when the Paper is very much inclined to it in liach
a manner as to keep always parallel to the Axis of the
Prifm, the whitenefs of the whole Light upon the
Paper will according to the inclination of the Paper
this way, or that way, change either into yellow and
red, as in the pofture de^ or into blue and violet, as
in the pofture ^s. And if the Light before it fall upon
the Paper be twice refraded the fame way by two pa-
rallel Prilms, thefe Colours will become the more con-
Ipicuous. Here all the middle parts of the broad beam
of white Light which fell upon the Paper, did without
any confine of Ihadow to modify it, become, coloured
all over with, one uniform Colour, the Colour being al-
ways the fame in the middle of the Paper as at the
edges, and this Colour changed according the various
obliquity of the rerie6fing Paper, without any change
in the refradions or fhadow, or in the Light which
fell upon the Paper. And therefore tlicfe Colours are

to.

X85]

to be derived from fome other caufe than the new mo-
difications of Light by refradions and fhadows.

If it be asked, What then is their caufe ? I anfwer,
That the Paper in the pofture de ^ being more ob-
lique to the more refrangible rays than to the lefs re-
frangible ones, is more ftrongly illuminated by the lat-
ter than by the former, and therefore the lefs refran-
gible rays are predominant in the reflected Light. And
wherever they are predominant in any Light they tinge
it with red or yellow, as may in fome meafure appear by
the firft Propofition of the firft Book,and will more fully
appear hereafter. And the contrary happens in the
poll:ure of the Paper e%, the more refrangible rays be-
ing then predominant which always tinge Light with
blues and violets.

EX PER. IV.

The Colours of Bubbles with which Children play
are various, and change their lituation varioufly, with-
out any refped to any confine of fhadow. If fuch a
Bubble be covered with a concave Glafs, to keep it from
being agitated by any wind or motion of the Air, the
Colours will ilowly and regularly change their fitua-
tion, even whilft the Eye, and the Bubble, and all Bo-
dies which emit any i^ight, or caft any fhadow, re-
main unmoved. And therefore their Colours arife from
fome regular caufe which depends not on any confine of
fliadow. What this caufe is will be fhewed in the next
Book..

[8(5]

To tlicfe Experiments may be added the tenth Ex-
periment of the firft Book, where the Sun's Light in a
dark Room being trajeded through the parallel luperfi-
cies of two Prifms tied together in the form of a Paral-
lelopide, became totally of one uniform yellow or red
Colour, at its emerging out of the Prifms. Here, in
the production of thefe Colours, the confine of fhadow
can have nothing to do. For the Light changes from
white to yellow,orange and red fucceffively,withoutany
alteration of the confine of (hadow: And at both edges of
the emerging Light where the contrary confines of Hia-
dow ought to produce different etfeds, the Colour is
one and the fame, whether it be white, yellow, orange
or red : And in the middle of the emerging Light,
where there is no confine of ihadow at all, the Colour
is the very fame as at the edges, the whole Light at its
very firft emergence being of one uniform Colour, whe-
ther white, yellow, orange or red, and going on thence
perpetually without any change of Colour, fuch as the
confine of fhadow is vulgarly fuppofed to work in re-
fracted Light after its emergence. Neither can thefe
Colours arife from any new modifications of the Light,
by refradtions, becaufe they change lucceffively from
white to yellow, orange and red, while the refradions
remain the fame, and alio becaufe the refractions are
made contrary ways by parallel fuperficies which de-
ftroy one anothers efteCts. They arife not therefore
from any modifications of Light made by refra61:ions
jind fhadows, but have fome other caufe. What that
caufe is we ihewed above in this tenth Experiment,
and need not here repeat it.

There

[87]

There is yet another material circumftance of this
Experiment. For this emerging Light being by a third Fig. 2 a.
Prifm HI K refradled towards the Paper PT, and there Tart i.
painting the ufual Colours of the Prifm, red, yellow,
green, blue, violet : If thefe Colours arofe from the
refractions of that Prifm modifying the Light, they
wonld not be in the Light before its incidence on that
Prifm. And yet in that Experiment v/e found that
when by turning the two firft Prifms about their com-
mon Axis all the Colours were made to vanilli but the
red 3 the Light which makes that red being left alone,
appeared of the very fame red Colour before its inci-
dence on the third Prifm. And in general we find by
other Experiments that when the rays which diifer in
refrangibility are feparated from one another, and any
one fort of them is confidered apart, the Colour of the
Light which they compofe cannot be changed by any
refradion or reflexion whatever, as it ought to be were
Colours nothing elfe than modifications of Light caufed
by refradions, and reflexions, and lliadows. This un-
changeablenefs of Colour I am now to defcribe in the
following Propofition.

PROP. II. THE OR. I L

u^ll homogeneal L'-ght has its frofer Colour anjisi>ertng to
its degree of refrangdiltty^ and that Colour cannot be
changed by rejlexions and refraHions,

In the Experiments of the 4th Propofition of the firft
Book, when I had feparated the Jieterogeneous rays
from one another, the Spectrum p t formed by the fepa-
rated

[88]

rated rays, did in the progrefs from its end p, on which
the moft refrangible rays fell, unto its other end t, on
which the leaft refrangible rays fell, appear tinged with
this Series of Colours, violet, indico, blue, green, yel-
low, orange, red, together with all their intermediate
degrees in a continual fucceflion perpetually varying :
So that there appeared as many degrees of Colours, as
there were forts of rajs differing in refrangibility.

EX PER. V.

Now that thefe Colours could not be changed by re-
fraction, I knew by refradting with a Prifm fometimes
one very little part of this Light, fometimes another
very little part, as is defcribed in the 1 2th Experiment
of the firft Book. For by this refraftion the Colour of
the Light was never changed in the leaft. If any part
of the red Light was refracted, it remained totally of
the fame red Colour as before. No orange, no yel-
low, no green, or blue, no other new Colour was pro-
duced by that refradiion. Neither did the Colour any
ways change by repeated refradions, but continued al-
ways the fame red entirely as at firft. The like con-
ftancy and im.mutability 1 found alio in the blue, green,
and other Colours. So alfo if 1 looked through a Prifm
upon any body illuminated with any part of this homo-
geneal Light, as in the 1 4-th Experiment of the firft
Book is defcribed ; I could not perceive any new Co-
lour generated this way. All Bodies illuminated with
compound Light appear through Prifms confufed ( as
was faid above) and tinged with various new Colours,
but thofe illuminated with homogeneal Light appeared

throug^h

[89]

through Prlfms neither kf^ diftinft, nor otherwife co-
loured, than when viewed with the naked Eyes. Their
Colours were not in the leaft changed by the refra£tion
of the inter]^ofed Prilm. 1 Ipeak here of a fenfible
change of Colour : For the Light which 1 liere call ho~
mogeneal, being not abfolutely homogeneal, there ought
to arife fome little change of Colour from its heteroge-
neity. But if that heterogeneity was fo little as it might
be made, by the laid Experiments of the fourth Propo'
fition, that change w^as not fenhble, and therefore, in
Experiments where fenfe is judge, ought to be accoun-
ted none at all.

EXPER. VI.

And as thcfe Colours were not changeable by refra-
ftions. To neither were they by reflexions. For all
white, grey, red, yellow, green, blue, violet Bodies, as
Paper, Afhes, red Lead, Orpiment, Indico, Bife, Gold,
Silver^ Copper, Grafs, blue Flowers, Violets, Bubbles
of Water tinged with various Colours, Peacock's Fea-
thers, the tincture of Lignum Nefhriticum^ and fuch
like, In red homogeneal Light appeared totally red, in
blue Light totally blue, in green Light totally green,
and fo of other Colours. In the homogeneal Light of
of any Colour they all appeared totally of that fame
Colour, with this only ditierence, that fome of them
refleded that Light more ftrongly, others more faintly.
I never yet found any Body which by reflecting homo-
geneal Light could fenfibly change its Colour.

M From

[90]

From all which it is manifeft, that if the Sun's Light
confiftcd of but one fort of rays, there would be but
one Colour in the whole World, nor would it be pof-
fible to produce any new Colour by reBexions and re-
fraftions, and by confequence that the variety of Co-
lours depends upon the compoiition of Light.

'DEFINIT ION.

The homogeneal light and rays which appear red,
or rather make Obje6ts appear ib, 1 call rubrific
or red'makng ; thofe which make Objects appear
yellow, green, blue and violet, 1 call yellow-ma-
king, green-making^ blue-making, violet-making,
and ib of the reft. And if at any time I fpeak of
light and rays as coloured or endued with Co-
lours, *I would be underftood to fpeak not philo-
fophically and properly, but grolly , and accor-
ding to fuch conceptions as vulgar People in fee-
ing all thefe Experiments would be apt to frame.
For the rays to fpeak properly are not coloured.
In them there is nothing elfe than a certain power
and difpofition to ftir up a lenfation of this or that
Colour. For as found in a Bell or mufical String,
or other founding Body, is nothing but a trem-
bling Motion, and in the Air nothing but that
Motion propagated from the Objedt, and- in the
Senforium *tis a fenfe of that Motion under the
form of found ; fo Colours in the Objed are no-
thing but a difpoiition to refled: this or that fort
of rays more copiouUy than the reft ; in the rays
they are nothing but their difpofitions to propa-
gate

[pt]

gate this or that Motion into the Senforium, and
in the Senforium they are lenlations of thofe Mo-
tions under the forms of Colours.

PROP. III. PROB. I.

To define the refravgibtltt'j of the jeveral forts of homo^
oeneal Lt<yht anfwermg to the feveral Colou'i

it J.

For determining this Problem 1 made the following
Experiment.

EXPER. VII.

When I had caufed the rectilinear line (ides A F, G M, JFtg. 4.
of the Speftrum of Colours made by the Prifm to be
diftindlly denned, as in the fifth Experiment of the
firft Book is defcribed, there were found in it all the
homogeneal Colours in the fame order and fituation
one among another as in the Spectrum of fimple Light,
defcribed in the fourth Experiment of that Book. For
the Circles of which the Sped:rum of compound Light
PT is compofed, and which in the middle parts of
the Sped: rum interfere and are intermixt with one ano-
ther, are not intermixt in their outmoft parts where
they touch thofe redilinear fides AF and GM. And
therefore in thofe redilinear fides when diftinftly defi-
ned, there is no new Colour generated by refraftion. I
obferved alfo, that if any where between the two out-
mofi: Circles TMF and PC A a right line, as 7^, was
crofs to the Spectrum, fo as at both ends to fall per-
pendicularly upon its reftilinear fides, there appeared

M 2 one

1^ 92 ]

one and the ratne Colour and degree ot Colour from one-
end of this line to the other. I delineated therefore in
a Paper the perimeter of the Spectrum FA PGMT,
and in trying the third Experiment of the hrft Book, I
held the Paper lb that the Spedtrum might fall upon
this delineated Figure, and agree with it exactly, whilft
an Affiftant whofe Eyes for dilKnguiiliing Colours were
more critical than mine, did by right lines ^3) -c^, (^^crc
drawn crofs the Spedtrum, note the confines of the Co-
lours that is of the red M*/3F of the orange ayc/>/i^ of
the yellow y s ^^, of the green '- 1 s ^ , of the blue n » x 9 ,
of the indico tXM>i5 and of the violet xGAm. And
this operation being divers times repeated both in the
lame and in leveral Papers , I found that the Ob-
fervations agreed well enough with one another, and
that the redtiiinear fides M G and FA were by the faid
crofs lines divided after the manner of a mufical Chord.
Let GM be produced to X, that MX may be equal
toGM, and conceive GX, xX, 'X, ^'X,,^X, yX, «Xy
MX, to be in proportion to one another, as the num-
bers I, 9-, 6, 4^ p 1' ?6' i' and lb to rcprelent the.
Chords of the Key, and of a Tone, a third Minor, a
fourth, a fifth, a fixth Major, a feventh, and an eighth
above that Key : And the intervals M -^ , " 7 , 7 - , ^ « , 1 ',
'^, and xG, will be the fpaces which the fe vera I Co-
lours ( rcd^ orange, yellow, green, blue, indico, violet )
take up.

Now thefe inter\als or fpaces fubtending the diffe-
rences of the refractions of the rays going to the limits.
of thofe Colours, that is^ to the points M, a, 7, =, 15,/, x, G,
may without any fenfiblc Etror be. accounted propor-
tional to the differences of the fines of reiradtion of thofe

rays

rays having one common fine of incidence, and there-
fore fince the common fine of incidence of the moft and
lea ft refrangible rays out of Glafs into Air was, (by a
method defcribed above ) found in proportion to their
fines of refradion, as 50 to 77 and 78, divide the dif-
ference between the fines of refraction 77 and 78, as the
line G M is divided by thofc intervals, you will have

77. 77«-> 77'-' 77v 77i^ 77l' 77'.,, 7^, the fines of
refradion of thole rays out of Glafs into Air ,. their
common fine of incidence being 50. So then the fines
of the incidences of all the red-making rays out of
Glafs into Air, were to the fines of their refraftions,
not greater than 50 to 77, nor lefs than 50 to 77«-, but
varied from one another according to all interme-
diate Proportions. And the fines of the incidences
of the green-making rays were to the fines of
their refractions in all proportions from that of 50
to 77^, unto that of 50 to 77-;. And by the like. limits
above-mentioned were the refradions of the rays be-:
longing to the reft of the Colours defined, the fines of
the red- making rays extending from 77 to 778-, thofe
of the orangcrmaking from 775 to 77^ j thofe of the yel-
low-making from 77^ to 77 1, thofe of the green-making
from 777 to 7 7x J thofe of the blue-making from 77^ to
775, thofe of the indico-making from 77-j to 77,;, and •
thofe of the violet from 77^ to 78.

Thefe are the Laws of the refrad ions made out of
Glafs into Air, and thence by the three Axioms of tlie
hrft Book the Laws of the refractions made out of Air.
into Glafs areeafily derived.:

EXPER.

[94]

EX PER. VIII.

I found moreover that when Light goes out of Air
through feveral contiguous refrafting Mediums as
through Water and Glafs, and thence goes out again
into Air, whether the refracting fupcrficies be parallel
or inclined to one another, that Light as often as by
contrary refractions 'tis fo corrected, that it emergeth
in hues parallel to thofe in which it was incident,
continues ever after to be white. But if the emer-
gent rays be inclined to the incident, the whitenefs of
the emerging Light will by degrees in palling on from
the place of emergence, become tinged in its edges with
Colours. This I tryed by refrading Light with Prifms
of Glafs within a prifmatick Veffel of Water. Now thofe
Colours argue a diverging and feparatioii of the hetero-
geneous rays from one another by means of their un-
equal refradions, as in what follows will more fully
appear. And, on the contrary, the permanent white-
nets argues, that in like incidences of the rays there is
no fuch leparation of the emerging rays, and by confe-
quence no inequality of their whole refradions. Whence
1 ieem to gether the two following Theorems.

I. The Exceffes of the fines of refraction of feveral
forts of rays above their common fine of incidence when
the refractions are made out of divers denfer mediums
immediately into one and the fame rarer medium, are
to one another in a given Proportion.

a. The

[P5]

a. The Proportion of the line of incidence to the fine
of refraction of one and the fame fort of rays out of one
medium into another, is compofed of the Proportion of
the line of incidence to the line of refradion out of the
firfl: medium into any third medium, and of the Pro-
portion of the line of incidence to the line of refradion
out of that third medium into the fecond medium.

By the firft Theorem the refractions of the rays of
every fort made out of any medium into Air are known
by having the refraClion of the rays of any one fort. As
for inllance, if the refractions of the rays of every fort
out of Rain-v/ater into Air be delired, let the common
fine of incidence out of Glafs into Air be fubduded
from the lines of refraCtion, and the Excefl'es will be
ay, i-j\y if-, 27^ > 27-;, I-]], 279-, 28. Suppofenow
that the fine of incidence of the leaft refrangible rays be
to their fine of refraCtion out of Rain-water into Air as
three to four, and fay as i the ditference of thofe fines
is to 5 the fine of incidence, fo is 27 the leaft of the
Excelies above-mentioned to a fourth number 8 1 ; and
81 will be the common fign of incidence out of Rain-
water into Air, to which line if you add all the above-
mentioned Excefles you will have the defired fines of
the refractions 108, loSs, 1087, 1087 ^ io8i, loSf,
1089, 109.

By the latter Theorem the refraCtion out of one me-
dium into another is gathered as often as you have
the refractions out of them both into any third medium.
As if the fine of incidence of any ray out of Glafs into
Air be to its fine of refraCtion as ao to ^ i, and the fine
of incidence of the fame ray out of Air into Water, be

to its fine of refraftion as four to three ; the fine of
incidence of that ray out of Glafs into Water will be to
its fine of refraction as lo to ^ i and 4 to^ joyntly, that
is, as tjie Fadum of ao and 4. to the Factum of 3 1 and
3, or as 80 to 93.

And thele Theorems being admitted into Opticks,
there would be fcope enough of handling that Science
voluminoufly after a new manner ; not only by teaching
thofe things which tend to the perfettion of vilion, but
alfo by determining mathematically all kinds of Phaeno-
mena of Colours which could be produced by refra-
dtions. For to do this, there is nothing elfe requifite
than to find out the reparations of heterogeneous rays,
and their various mixtures and proportions in every
mixture. By this way of arguing 1 invented almoft
all the Phsenomena defcribed in thefe Books, befide fome
others lefs neceflary to the Argument ; and by the
fucceffes I met with in the tryals, I dare promife, that
to him who iTiall argue truly, and then try all things
with good Glafles and fufficient circumfpection, the
expeded event will not be wanting. But he is firft to
know what Colours will arife from any others mixt iu
any affigned Proportion,

PROP. IV. THEOR. IIL

Colours mm he produced iy compofition which /ball ht like
to the Colours of homogeneal Ljo^ht as to the affenrame
of Colour^ but not as to the immuta/nlity of Colour and
conjlttution of Light. j4nd thofe Colours ^y ho-w much
they are more compounded b'j jo much are they UJs fuU
■ and inteufe^ and by too much comfo/ition they may be

diluted

[97]

diluted attd 'weakened till they ceaje. i here nia\' be
alfo Colours froduced b'j comfofitioyi^ 'which are not jitlh
like an'j oj the Colours of hsmogeneod Light.

For a mixture of homogeneal red and yellow com-
pounds an orange, like in appearance of Colour to that
orange which in the feries of unmixed prifmatick Co-
lours lies between them; but the Light of one orange
is homogeneal as to refrangibility, that of the other is
heterogeneal, and the Colour of the one , if viewed
through a Prifm, remains unchanged, that of the other
is changed and refolved into its component Colours red
and yellow. And after the fame manner other neigh-
bouring homogeneal Colours may compound new Co-
lours, like the intermediate homogeneal ones, as yel-
low and green, the Colour between them both, and af-
terwards, if blue be added, there w^ll be made a green
the middle Colour of the three which enter the com.po-
lition. For the yellow and blue on either hand,if they are
equal in quantity they draw the intermediate green equal-
ly towards themfelves in compofition, and fo keep it as
it were in equillbrio, that it verge not more to the
yellow on the one hand, than to the blue on the other,
but by their mixt ad:ions remain ftili a middle Colour.
To this mixed green there may be further added
fome red and violet, and yet the green will not prefent-
ly ceafe but only grow lefs full and vivid, and by in-
creaiing the red and violet it will grow more and more
dilute, until by the prevalence of the added Colours it
be overcome and turned into whitenefs, or fome other
Colour. So if to the Colour of any homogeneal Light,
the Sun's white Light compofed of all lorts of. rays be

N added,

added, tltat Colour will not vanifh or change its fpe-
cies but be diluted, and by adding more and more white
it will be diluted more and more perpetually. Laft-
ly, if red and violet be mingled, there will be generated
according to their various Proportions various Purples,
fuch as are not like in appearance to the Colour ot any
homogeneal Light, and of theie Purples mixt with yel-
low and blue may be made other new Colours.

PROP. V. THEOR. IV.

Whitenefs and all gre>j Colours Set'ween 'white and Mach^
ma'j be. compounded o\ Colours^ and the isuhitenefs of the
Suns Light is compounded of all the f?imar>^ Colow's
mixt in a due pofortion.

The Proof hy Experiments.

EX PER. IX.

jr^tr. e. The Sun fhining into a dark Chamber through a
little round Hole in the Window fhut, and his Light
being there refraded by a Prifm to call his coloured
Image P T upon the oppohte Wall : I held a white Pa-
per V to that Image in fuch m^anner that it might be
illuminated by the coloured Light retiected from thence,
and yet not intercept any part of that Light in its paf-
fage from the Prifm to the Spedrum. And I found that
when the Paper was held nearer to any Colour than to
the reft, it appeared of that Colour to which it ap-
proached nearcft 3 but when it was equally or almoft

equally

199}

equally diftant from all the Colours, Co that it might
be equally illuiriinated by them all it appeared white.
And in this laft lituation of the Paper, if fome Colours
were intercepted, the Paper loll its white Colour, and
appeared of the Colour of the reft of the Light which
was not intercepted. So then the Paper was illuminated
with Lights of various Colours, namely, red, yellow,
green, blue and violet, and every part of the Light re-
tained its proper Colour, until it was incident on the
Paper, and became retiefted thence to the Eye ; fo that
if it had been either alone (the reft of the Light being
intercepted) or if it had abounded moft and been pre-
dominant in the Light retleded from thePaper,it would
have tinged the Paper w^ith its own Colour ; and yet be-
ing mixed wdth the reft of the Colours in a due propor-
tion, it made the Paper look white, and therefore by a
compofition with the reft produced that Colour. The
feveral parts of the coloured Light reflected from the
Spedf rum, whilft they are propagated from thence thro'
the Air, do perpetually retain their proper Colours,
becaufe wherever they fall upon the Eyes of any Specta-
tor, they make the feveral parts of the Spedrum to
appear under their proper Colours. They retain there-
lore their proper Colours when they fall upon the Pa-
per V, and lb by the confufion and perfed mixture of
thole Colours compound the whitenefs of the Light
reflected from thence.

EX PER. X.

Let that Spedrum or folar Lnage P T fall now upon Ftg. 6.
the Lens M N above four Inches broad, and about fix

N 2 Feet

[ loo]

Feet diftant from the Piifm ABC, and fo figured that
it may caufe the coloured Light which divergeth from
the Prifm to converge and meet again at its Focus G,
about fix or eight Feet diftant from the Lens, and
tliere to fall perpendicularly upon a white Paper DE.
And if you move this Paper to and fro, you will per-
ceive that near the Lens, as at de^ the whole folar Image
(fuppofe at pt) will appear upon it intenfly coloured
after the manner above-explained, and that by receding^
from the Lens thofe Colours will perpetually come to-
wards one another, and by mixing more and more di-
lute one another continually, until at length the Paper
come to the Focus G, where by a perfed mixture they
will wholly vanifli and be converted into whitenefs, the
whole Light appearing now upon the Paper like a little
white Circle. And afterwards by receding further from
the Lens, the rays which before converged will now
crofs one another in the Focus G, and diverge from
thence, and thereby make the Colours to appear again,
but yet in a contrary order ; fuppofe at c^£ , where the
red t is now above which before was below, and the
violet p is below which before v^-as above.

Let us now ftop the Paper at the Focus G where
the Light appears totally white and circular, and let us
Gonfider its whitenefs. I fay, that this is compofed of
the converging Colours. For if any of thofe Colours
be intercepted at the Lens, the whitenefs will ceafe and
degenerate into that Colour which arifeth from the
compofition of the other Colours which are not inter-
cepted. And then if the intercepted Colours be let
pafs and fall upon that compound Colour, they mix
with it, and by their mixture rcttore the whitenefs.

So.

C lOI ]

So if the violet, blue and green be intercepted, the re-
maining yellow, orange and red will compound upon
the Paper an orange, and then if the intercepted Co-
lours be let pafs they will fall upon this compounded
orange, and together with it decompound a white. So
alio if the red and violet be intercepted, the remaining
yellow, green and blue, will compound a green upon
the Paper, and then the red and violet being let pafs
will fall upon this green, and together with it decom-
pound a white. And that in this compoiition of white
the feveral rays do not fufFer any change in their colori-
fic qualities by ading upon one another, but are only
mixed, and by a mixture of their Colours produce
white, may further appear by thefe Arguments.

If the Paper be placed beyond the Focus G, fuppofe
at o'f , and then the red Colour at the Lens be alternate-
ly intercepted, and let pafs again, the violet Colour on
the Paper will not futfer any change thereby, as it ought
to do if the feveral forts of raysaded upon one another
in the Focus G, where they crofs. Neither will the
red upon the Paper be changed by any alternate flop-
ping, and letting pafs the violet which crolTeth it.

And if the Paper be placed at the Focus G, and the
white round Image at G be viewed through the Prifm
HIK, and by the refradion of that Prifm be tranflated
to the place rv, and there appear tinged with various
Colours, namely, the violet at v and red au r , and
others between, and then the red Colour at the Lens be
often ftopt and let pafs by turns, the red at r will ac-
cordingly difappear and return as often, but the violet;
at V will not thereby fuifer any change. And lb by
flopping and letting pafs alternately the blue at the

Lens.

[ 102 ]

Lens, the blue at r will accordingly dilappear and re-
turn, withoutany change made in the red at r. The
red therefore depends on one ibrt of rays, and the blue
on another fort, which in the Focus G where they are
commixt do not aft on one another. And there is the
lame realbn of the other Colours.

I conlidered further, that when the moft refrangible
rays Pp, and the leaft refrangible ones Tt, are by con-
verging inclined to one another, the Paper, if held very
oblique to thofe rays in the Focus G, might relieft one
Ibrt of them more copioufly than the other Ibrt, and by
that means the refledted Light would be tinged in that
Focus with the Colour of the predominant rays, pro-
vided thofe rays feverally retained their Colours or co-
lorific qualities in the compolition of white made by
them in that Focus. But if they did not retain them
in that white, but became all of them feverally endued
there with a difpofition to ftrike the fenfe with the per-
ception of white, then they could never lofe their white-
neis by fuch reflexions. I inclined therefore the Paper
to the rays very obliquely, as in the fecond Experiment
of this Book, that the moft refrangible rays might be
more copioufly reflected than the reft, and the white- ^
nets at length changed lucceflively into blue, indico^
and violet. Then 1 inclined it the contrary way, that
the moft refrangible rays might be more copious in the
refleded Light than the reft, and the whitenefs turned
fucceflively to yeUow, orange and red.

Laftly, I made an Inftrument XY in fafhion of a
Comb, whofe Teeth being in num.ber lixteen were
about an Inch and an half broad, and the intervals of the
Teeth about two Inches wide. Then by interpoflng

fuc-

[103]

fucceffively the Teeth of this Inftrumcnt near the Lerrs^
I intercepted part of the Colours by the interpofcd
Tooth, whilft the reft of them went on through the in-
terval of the Teeth to the Paper D E, and there pain-
ted a round folar Image. But the Paper I had firft pla-
ced fo, that the Image might appear white as often
as the Comb was taken away ; and then the Comb be-
ing as was laid interpofed, that whitenefs by reafon of
the intercepted part of the Colours at the Lens did al-
ways change into the Colour compounded of thofe
Colours which were not intercepted, and that Colour
was by the motion of the Comb perpetually varied fo,
that in the palling of every Tooth over the Lens all
thefe Colours red, yellow, green, blue and purple, did
always fucceed one another. I caufed therefore all the
Teeth to pais fucceffively over the Lens, and when the
motion was flow, there appeared a perpetual fucceffion
of the Colours upon the Paper : But if I fo much acce-
lerated the motion, that the Colours by reafon of their
quick fucceffion could not be diftinguilhed from one
another, the appearance of the iingle Colours ceafed.
There was no red, no yellow, no green, no blue, nor
purple to be feen any longer, but from a confulion of
them all there arofe one uniform white Colour. Of tj^e
Light which now by the mixture of all the Colours ap-
peared white, there was no part really white. One
part was red, another yellow, a third green, a fourth
blue, a fifth purple, and every part retains its proper
Colour till it ftrike the Senforium. If the impreffions
follow one another llowly, fo that they may be ieve-
raliy perceived, there is made a diftind: fenfation of all
the Colours one after another in a continual fucceffion.

But

[104]

But If the impreflions follow one another lb quickly
that they cannot be feverally perceived, there arifeth
out of them all one common fenlation, which is nei-
ther of this Colour alone nor of tliat alone, but hath it
lelf indifferently to 'em all, and this is a lenfation of
whitenefs. By the quicknefs of the fucceffions the im-
preffions of the feveral Colours are confounded in the
Senforium, and out of that confution arileth a mixt icn-
fation. If a burning Coal be nimbly moved round in a
Circle with Gyrations continually repeated, the whole
Circle will appear like hre ; the reafon of wiiich is, that
the fenfation of the Coal in the feveral places of that
Circle remains impreft on the Senforium, until the
Coal return again to the fame place. And lb in a
quick confecution of the Colours the impreffion of every
Colour remains in the Senforium, until a revolution of
all the Colours be compleated, and that firft Colour re-
turn again. The impreffions therefore of all the fucceffive
Colours are at once in theSenlbrium,and joyntly ftir up
a fenfation of them all ; and lb it is manifeft by this Ex-
periment, that the commixt impreffions of all the Co-
lours do hir up and beget a feniation of white, that is,
that whitenefs is compounded of all the Colours. j

And if the Comb be now taken away, that all the
Colours may at once pals from the Lens to the Paper,
and be there intermixed, and together relie<5ted thence
to the Speftators Eyes ; their impreffions on the Senfo-
rium bemg now more fubtily and perfedly commixed
there, ought much more to iHr up a fenfation of white-
•iiefs.

You

C 105 ]

You may inftead of the Lens ufe two Fril'ms HI K
andLMN, which by refractmg the coloured Light
the contrary way to that of the firft refraction, may
make the diverging rays converge and meet again in G,
as you fee it reprefented in tlie feventh Figure. For Ftg. 7.
where they meet and mix they will compote a white
Light as when a Lens is uied.

EX PER. XL

Let tl\e Sun's coloured Image PT fall upon the Wall Fig- 8.
of a dark Chamber, as in the third Experiment of the
lirftBook, and let the fame be viewed through a Prifm
a be, held parallel to the Prifm ABC, by whofe refra-
dion that Image was made, and let it now appear lower
than before, fuppofe in the place S over againll the red
colour T. And if you go near to the Image PT, the
Speftrum S will appear oblong and coloured like the
image PT; but if you recede from it, the Colours of
the Spedrum S will be contracted more and more, and
at length vanilli, that SpeCtrum S becoming perfectly
\ round and v/hite ; and if you recede yet further, the
^Colours will emerge again, but in a contrary order.
Now that Speftrum S appears white in that cafe when
the rays of feveral forts which converge from the feve*
ral parts of the Image PT, to the Prifm a be, are fo
refracted unequally by it, that in their paffage from the
Prifm to the Eye they may diverge from one and the
iame point of the Spectrum S, and fo fall afterwards
upon one and the fame point in the bottom of the Eye,
mid there be min2,led.

O And

[lod]

And further, if the Comb be here made ufe of, by
whofe Teeth the Colours at the Image PT may be luc-
ceffively intercepted ; the Spedrum S when the Comb
is moved flowly will be perpetually tinged with iiic-
ceflive Colours : But when by accelerating the motion
of the Comb, the fucceffion of the Colours is fo quick
that they cannot be feverally feen, that Spectrum S, by
a confufed and mixt lenlation of them all, will appear
white.

EXPER. XII.

Jpio: 9. The Sun fhining through a large Prifm ABC upon
a Comb X Y, placed immediately behind the Prifm, his
Light which paffed through the interlaces of the Teeth
fell upon a white Paper DE. The breadths of the
Teeth were equal to their interftices, and feven Teeth
together with their interftices took up an Inch in
breadth. Now w^hen the Paper was about two or
three Inches diftant from the Comb, the Light which
paffed through its feveral interftices painted ib many
ranges of Colours kl, mn, op, qr, ^r. which were
parallel to one another and contiguous, and without anyi
mixture of white. And thefe ranges of Colours, if the
Comb was moved continually up and down with a re-
ciprocal motion, afcended and defcended in the Paper,^
and when the motion of the Comb was fo quick, that
the Colours could not be diftinguifhed from one another,
the whole Paper by their confulion and mixture in the
Senforium appeared white.

Let

[ 107 ]

Let the Comb now reft, and let the Paper be remo-
ved further from the Prifm, and the feveral ranges of
Colours will be dilated and expanded into one another
more and more, and by mixing their Colours will di-
lute one another, and at length, when the diftance
of the Paper from the Comb is about a Foot , or a
little more ( fuppofe in the place i D 2 E ) they will
lb far dilute one another as to become white.

With any Obftacle let all the Light be now ftopt
which paffes through any one interval of the Teeth, ib
that the range of Colours which comes from thence may
be taken away, and you will fee the Light of the reft of
the ranges to be expanded into the place of the range
taken away, and there to be coloured. Let the inter-
cepted range pafs on as before, and its Colours falling
upon the Colours of the other ranges, and mixing with
them, will reftore the whitenefs.

Let the Paper 2D 2 E be now very much inclined to
the rays, fo that the moft refrangible rays may be more
copioufly reflefted than the reft, and the white Colour
of the Paper through the excefs of thofe rays will be
•v changed into blue and violet. Let the Paper be as
\nuch inclined the contrary way, that the leaft refran-
gible rays may be now more copioufly refleded than
the reft, and by their excefs the whitenefs will be
changed into yellow and red. The feveral rays there-
fore in that white Light do retain their colorific qua-
lities, by which thole of any fort, when-ever they be-
come more copious than the reft, do by their excefs
and predominance cauie their proper Colour to ap-
pear.

O 5 And

[io8]

And by the fame way of arguing, applied to the third
Experiment of this Book, it may be concluded, that
the white Colour of all refracted Light at its very firll:
emergence, where it appears as white as before its inci-
dence^ is compounded of various Colours.

EX PER. XIII.

In the foregoing Experiment the feveral intervals of
the Teeth of the Comb do the office of fo many Prifms,
every interval producing the Phaenomenon of one Prifm.
Whence inftead of thofe intervals ufing feveral Prifm?, I
try'd to compound whitenefs by mixing their Colours,and
did it by ufing only three Prifms, as alfo by ufing only
Fig' lo- two as follows. Let two Prifms ABC and a b c, whole
refrading Angles B and b are equal,be fo placed parallel
to one another, that the refrafting Angle B of the one
may touch the Angle c at the bale of the other, and
their planes CB and cb, at which the rays emerge, may
lye in directum. Then let the Light traje^ed through
them fall upon the Paper M N, dillant about 8 or i 2 /
Inches from the Prifms. And the Colours generatedr
by the interior limits B and. c of the two Prifms, wil(
be mingled at PT, and there compound white. For if
either Prifm be taken away, the Colours made by the
other will appear in that place PT, and when the Prifm
is reftoredto its place again, fo that its Colours may
there tall upon the Colours of the other, the mixture
of them both will reftore the whitenels.

This

This Experiment fucceeds alio, as I have tryed, when
the Angle b of the lower Prilm, is a little greater than
the Angle B of the upper , and between the interior
Angles B and c, there intercedes ibme fp:ice B c, as is
teprelented in the Figure, and the retracing planes
BC and be, are neither in directum, nor parallel to
one another. For there is nothing raore requifite to
the fucceis of this Experiment, than that the rays of all
forts may be uniformly mixed upon the Paper in the
place PT. If the moft refrangible rays coming from
the fuperiorPriim take up all theipace from M to P, the
rays of the fame fort which come from the inferior
Prifm ought to begin at P, and take up all the reft of the
fpacefrom thence towards N. If the leaft refrangible
rays coming from the fuperior Prifm take up the ipace
MT, the rays of the lame kind which come from the
other Prifm ought to begin atT, and take up the remaiir-
ing fpace T N. If one Ibrt of the rays which have in-
termediate degrees of refrangibility,, and come from the
fuperior Prifm be extended through the ipace MQ-, and
another ibrt of thofe rays through the fpace MR, and
a third fort of them through the ipace MS, the fame
forts of rays coming from the lower Prifm, ought to ilr
iluminate the remaining ipaces Q.N, RN, SN refpe-
dively. And the lame is to be underllood of all the
other ibrts of rays. For thus the rays of every fort wdll
be fcattered uniformly and evenly through the whole
fpace MN, and lb being every wiiere mixt in the fame
proportion, they muft every where produce the lame
Colour. And therefore lince by this mixture.they pro-
duce white in the exterior Ipaces M P and TN, they
muft alfo produce white in the interior ipace P T. This

is.

[no]

IS the reafon of the compofition by which whitcncfs
was produced in this Experiment, and by what other
way Ibever 1 made the like compofition the refult was
whitcnefs.

Laftly, If with the Teeth of a Comb of a due (ize,
the coloured Lights of the two Prifms which fall upon
the fpace PT be alternately intercepted, that fpace
PT, when the motion of the Comb is flow, will always
appear coloured, but by accelerating the motion of
the Comb fo much, that the fucceffive Colours can-
not be diftinguiflied from one another, it will appear
white.

EXPER. XIV.

Hitherto I have produced whitenefs by mixing the
Colours of Prifms. If now the Colours of natural Bo-
dies are to be mingled, let Water a little thickned with
Soap be agitated to raife a froth, and after that froth
has Hood a little, there will appear to one that fliall
view it intently various Colours every where in the
furfaces of the feveral Bubbles ; but to one that fhall
go fo far off that he cannot diftinguifh the Colours from
one another, the whole froth will grow white with a'
perfed whitenefs.

EXPER. XV.

Laftly, in attempting to compound a white by mixing
the coloured Powders which Painters ufe, 1 confidered
that all coloured Powders do fupprefs and ftop in
them a very coniiderable part of the Light by which

they

[Ill]

they are illuminated. For they become coloured by
reflecting the Light of their own Colours more copioufly,
and that of all other Colours morefparingly, and yet
they do not reflect the Light of their own Colours lb
copioufly as white Bodies do. If red Lead, for inftance,
and a white Paper, be placed in the red Light of the
coloured Spectrum made in a dark Chamber by the re-
fradion of a Prifm, as is defcribed in the third Eperi-
mentofthe firft Book 3 the Paper will appear more lu-
cid than the red Lead, and therefore refieds the red^-
making rays more copioufly than red Lead doth. And
if they be held in the Light of any other Colour, the
Light reflected by the Paper will exceed the Light re-
fie&ed by the red Lead in a much greater proportion.
And the like happens in Powders of other Colours.
And therefore by mixing fuch Powders we are not to
expert a fl:rong and fuU white, fuch as is that of Paper,
but fome dusky obfcure one, fuch as might arife from a
mixture of light and darknefs, or from white and black,
that is, a grey, or dun, or ruffet brown, fuch as are the
Colours of a Man's Nail, ofaMoufe, of Aflies, of or-
\dinary Stones, of Mortar, of Duft and Dirt in High-
\ways, and the like. And fuch a dark white I have
often produced by mixing coloured Powders. For thus
one part of red Lead,and Ave parts of Viride jEris^com-
pofed a dun Colour like that of a Moufe. For thele
tv\Ao Colours were feverally fo compounded of others,
that in both together were a mixture of all Colours ; and
there was lefs red Lead ufed than Vtride y^rw, becaufe
of the fulneis of its Colour. Again, one part of red
Lead, and four parts of blue Bife, compofed a dun Co-
lour verging a little to purple, and by adding to this a

certain

[112]

certain mixture of Orpiment and Vtnd't j!Eris in a due
proportion, the mixture loft its purple tincture, and be-
came perfedly dun. But the Experiment lucceeded beft
without Minium thus. To Orpiment I added by little
and little a certain full bright purple, which Painters
ufe until the Orpiment ceafed to be yellow, and became
ofa pale red. Then I diluted that red by adding a
little Viride ^>m, and a little more blue Bile than J/^i-
riiU jEris^ until it became of fuch a grey or pale white,
as verged to no one of the Colours more than to ano-
ther. For thus it became of a Colour equal in white-
neis to that of Afhes or of Wood newly cut, or of a
Man's Skin. The Orpiment refteded more Light than
did any other of the Powders, and therefore conduced
more to the whitenefs of the compounded Colour than
they. To affign the proportions accurately may be
difficult, by reaibn of the different goodneis of Pow-
ders of the lame kind. Accordingly as the Colour of
any Powder is more or lets full and luminous, it ought
to be ufed in a lefs or greater proportion.

Now confidering that thele grey and dun Colours
may be alfo produced by mixing whites and blacks, and
by conlequence differ from perfeft whites not in Species/
of Colours but only in degree of luminouiheis, it is ma-
. nifeft that there is nothing more requiiite to miake
them perfectly white than to increafe their Light fuffi-
ciently ; and, on the contrary, if by increahng their
Light they can be brought to perfect whitenefs, it will
thence alio follow, that they are of the lame Species of
Colour with the beft whites, and dift'er from them only
in the quantity of Light. And this 1 tryed as follows.
I took the third of the above-mentioned grey mixtures

(that

[113]

(^that which was compounded of Oipiment, Purple,
Bife and Viride Alerts) and rubbed it thickly upon the
lioor of my Chamber, where the Sun Ihone upon it
through the opened Caiement ; and by it, in the fha-
dow, 1 laid a piece of white Paper of the fame bignefs.
Then going from them to the dillance of 1 1 or 1 8 Feet,
fo that I could not difcern the unevennefs of the furface
of the Powder, nor the little fnadows let fall from the
gritty particles thereof ; the Powder appeared intenfly
white, fo as to tranfcend even the Paper it felf in white-
nefs, efpecially if the Paper were a little iTiaded from
the Light of the Clouds, and then the Paper compared
with the Powder appeared of fuch a grey Colour as the
Powder had done before. But by laying the Paper
where the Sun fhines through the Glafs of the Window,
or by Ihutting the Window that the Sun might fhine
through the Glafs upon the Powder, and by fuch other
£t means of increafing or decrealing the Lights where-
with the Powder and Paper were illuminated , the
Light wherewith the Powder is illuminated may be
made ftronger in fuch a due proportion than the Light

^ wherewith the Paper is illuminated, that they fhall both
appear exadly alike in whitenefs. For wdien I was
trying this, a Friend coming to viht me, I ftopt him
at the door, and before 1 told him what the Colours
were, or what I was doing ; I askt him. Which of the
two whites were the beft, and wherein they differed ?
And after he had at that diftance viewed them, well, he
anfwered, That they were both good wdiites, and that
he could not fay which was beft, nor wherein their Co-
lours differed. Now if you confider, that this white
of the Powder in the Sun-fhine was compounded of the

P Colours

[11+]

Colours which the component Powders ( Oipiment,
Purple, Bile, and Viride jEris) have in the fame Sun-
fhine, you muft acknowledge by this Experiment, as
well as by the former, that perfect whitenefs may be
compounded of Colours.

From what has been faid it is alfo evident, that the
whitenefs of the Sun's Light is compounded of all the
Colours wherewith the feveral forts of rays whereof
that Light coniifts, when by their feveral refrangibili-
ties they are feparated from one another, do tinge Paper
or any other white Body whereon they fall. For thole
Colours by Prop. i. are unchangeable, and whenever
all thofe rays with thole their Colours^ are mixt again^
they reproduce the fame white Light as before.

PROP. VI PROS. IL

Jn a mixture of ffimary Colours^ the quantity and quality
of each being given ^ to.knovj the Colour of the com--
founds

JFig.i I . With the Center O and Radius O D dcfcribe a Circle
ADF, and diftinguifli its circumference into feven parts/
D E, E F, F G, G A, A B, B C, CD, proportional to'
the feven muikal Tones or Intervals of the eight Sounds,
*SW, la^ ftty jol^ la^ miy fa^ jol^ contained in an Eight,
that is, proportional to the numbers ; , 7,, -f^, J-, 7,, -f^,
;. Let the hrft part D E reprefcnt a. red Colour, the
lecond E F orange, the third F G- yellow^ the fourth
GH green, the fifth AB blue, the lixth BC indico,
and the feventh CD violet. And conceive that thefe
are all the Colours of uncompoundcd Light gradually

palling

C "5 ]

pafling into one another, as they do when made by
Prilms ; the circumference DK FGABCD, repreien-
ting the whole leries of Colours from one end of the
Sun's coloured Image to the other, fo that from D to E
be all degrees of red, at E the mean Colour between red
and orange, from E to F all degrees of orange, at F the
mean between orange and yellow, from F to G all de^
grees of yellow, and lb on. Let p be the center of
gravity of the Arch DE, and q, r, s, t, v, x, the centers
of gravity of the Arches EF, EG, G A, A B, BC
and C D refpeftively, and about thole centers of gra-
vity let Circles proportional to the number of rays of
each Colour in the given mixture be defcribed; that is,
the circle p proportional to the number of the red-ma-
king rays in the mixture, the Circle q proportional to
the number of the orange-making rays in the mixture,
and fo of the reft. Find the common center of gravity
of all thole Circles p, q, r, s, t, v, x. Let that center
be Z ; and fromi the center of the Circle A D F, through
Z to the circumference, drawing the right line O Y,
the place of the point Y in the circumference fliall Ihew
■\the Colour ariling from the compofttion of all the Co-
lours in the given mixture, and the line OZ fhall be
proportional to the fulnefs or intenfenefs of the Colour,
that is, to its diftancc from whitenefs. As if Y fall in
the middle between F and G, the compounded Colour
iliall be the beft yellow ; if Y verge from the middle to-
wards F or G, the compounded Colour fhall according-
ly be a yellow, verging towards orange or green. IfZ
fall upon the circumference the Colour fliall be intenfe
.and florid in the higheft degree ; if it fall in the mid-
'way between the circumference and center, it fliall be

P 0. but

but halffo intenfe, that is, it Ihall be fiich a Colour as
would be made by diluting the intenleft yellow with an
equal quantity of whitenefs ; and if it fall upon the
center O, the Colour fliall have loft all its intcnfenels,
and become a white. But it is to be noted, That if the
point Z fall in or near the line O D, the main ingredients
being the red and violet, the Colour compounded (hall
not be any of the prifmatic Colours, but a purple, in-
dining to red or violet, accordingly as the point Z
lieth on the fide of the line DO towards E or towards C,
and in general the compounded violet is more bright and
more fiery than the uncompounded. Alfo if only two
of the primary Colours which in the Circle areoppofite
to one another be mixed in an equal proportion, the
point Z fhall fall upon the center O, and yet the Co-
lour compounded of thofe two Ihall not be perfedly
white, but fome flint anonymous Colour. For I could
never yet by mixing only two primary Colours produce
aperfed white. Whether it may be compounded of a
mixture of three taken at equal diftances in the circum-
ference I do not know, but of four or five I do not much
queftion but it may. But thefe are curiofities of little
or no moment to the underftanding the Phaenomena otV
nature. For in all whites produced by nature, there
ufes to be a mixture of all forts of rays, and by confe^
quence a compofition of all Colours.

To give an inftance of this Rule ; fuppofe a Colour is
compounded of thefe homogeneal Colours, of violet
1 part, of indico i part, of blue i parts, of green 3 parts,,
of yellow 5 parts, of orange 6 parts, and of red i o parts.
Proportional to thefe parts I defcribe the Circles x, v, t,
s, r, q, p refpe^tively, that is, fo that if the Circle x

[117] ^

be I, the Circle v may be i, the Circle t 2, the Circle
s ^, and the Circles r, qandp, 5, 6 and 10. Then I
find Z the common center of gravity of thefe Circles,
and through Z drawing the line O Y, the point Y falls
upon the circumference between E and F, fome thing
nearer to E than to F, and thence I conclude, that the
Colour compounded of thefe ingredients will be an
orange, verging a little more to red than to yellow.
Alfo 1 find that O Z is a little lefs than one half of
OY, and thence I conclude, that this orange hath a
little lefs than half the fulnefs or intenfenefs of an un-
compounded orange ; that is to lay, that it is fuch an
orange as may be made by mixing an homogeneal orange
with a good w^hite in the proportion of the line O Z to
the line Z Y, this proportion being not of the quantities
of mixed orange and white powders, but of the quan-
tities of the lis;hts relieved from them.

This Rule I conceive accurate enough for prailire,
though not mathematically acairate ; and the truth of
it may be fufficiently proved to fenfe, by flopping any
of the Colours at the Lens in the tenth Experiment of
this Book. For the reft of the Colours which are not
\ftopped, but pafs on to the Focus of the Lens, will
there compound either accurately or very nearly fuch
a Colour as by this Rule ought to refult from their
mixture.

PROPJ

Cu8j

PROP. VII. THEOR. V.

\^ll the Colours in the Univerje isjhich are made Sy Lj'^ht^
and defend 7iot on the fo^'joer of trnagmation^ are
either the Colours of homogeneal Lights^ or comfoanded
of thefe and that either accw ately or very nearly^ ac^
lordmg to the Kjule of the foregoing 'Problem,

For it has been proved ( in Prop.i. Li^.'i.) that the
changes of Colours made by retradions do not arife
from any. new modifications of the rays impreft by thofe
refractions, and by the various terminations of light
and (hadow, as has been the conftant and general opi-
nion of Philofophers. It has alfo been proved that the
feveral Colours of the homogeneal rays do conftantly
anfwer to their degrees of refrangibility, (Prop, i . Li^.i.
andProp.a. L^'/^.^.j and that their degrees of refrangi-
bility cannot be changed by refractions and retiexions,
{Vvop.2. Li^.^') and by confequence that thofe their
Colours are likewife immutable. It has alfo been pro-
ved diredlly by refracting and reflecting homogeneal
Lights apart, that their Colours cannot be changed,/
( Prop.i. LiLi.) It has been proved alio, that when'
the feveral forts of rays are mixed, and in crofling pafs
through the lame Ijpace, they do not aCt on one another
;ib as to change each others colorifick qualities, (Exper.
do. L.iLi.) but by mixing their aCtions in the Senfo-
rium beget a fenfation differing from what either would
do apart, that is a fenfation of a mean Colour between
-^hcir proper Colours ; and particularly when by the
Jioncourfe and mixtures .«f all forts of rays, a white

Colour

[119]

Colour Is produced, tlie white is a mixture of all tlie
Colours which the rays would have apart, ( Prop. 5.
Lik 1. ) The rays in that mixture do not lofe or alter
their feveral coloritick qualities, but by all their various
kinds of aftions mixt in the Senforium, beget a fenfa-
tion of a middling Colour between all their Colours
which is whitenefs. For whitenefs is a mean between
all Colours, having it felf indifterently to them all, fo
as with equal facility to be tinged vv'ith any of them.
A red Powder mixed with a little blue, or a blue with
a little red, doth not prefently lofe its Colour, but a
white Powder mixed with any Colour is prefently tin-
ged with that Colour, and is equally capable of beincr
tinged with any Colour what-ever. It has been fhewed
alfo, that as the Sun's Light is mixed of all forts of rays,
fo its whitenefs is a mixture of the Colours of all forts
of rays ; thofe rays having from the beginning their fe-
veral coloriiic qualities as well as their feveral refrangi-
bilities, and retaining them perpetually unchang'd not-
withft:mding any refradlions or reliexions they, may at
any time lliffer, and that when-ever any fort of the
\ Sun's rays is by any. means (as by reflexion in Exper. 9
vend 10. LiL i. or by refraction as happens in all re-^
fractions) feparated. from the reft, they then manifeft
their proper Colours. Thefe things have been proved,
ajid thefumof all this amounts to the Propolition here
to be proved. For if the Sun's Light is mixed of le-
veral forts of rays, each of which have originally their
feveral refrangibilities and colorifick quilities, and not-
withftanding their refractions and retiedtions, and their
various feparations or mixtures^ keep thofe their ori-
ginal properties perpetually the. lame without, altera-
tion ;

[I20]

tlon ; then all the Colours in the World mull: be fuch as
conllaiitly ought to arile from the original colorific qua-
lities of the rays whereof the Lights conlift by which
thole Colours are feen. And therefore if thereafon of
kny Colour what-ever be required, we have nothing elie
to do then to conlider how the rays in the Sun's Light
have by reflexions or refractions, or other caufes been par-
ted from one another ,or mixed together; or otherwile to
find out wliat forts of rays are in the Light by which
that Colour is made, and in what proportion ; and
then by the laft Problem to learn the Colour which
ought to arife by mixing thofe rays (or their Colours)
in that proportion. I fpeak here of Colours fo far as
they arife from Light. For they appear fometimes by
other caufes, as when by the power of phantafy we
fee Colours in a Dream, or a mad Man fees things before
him which are not there 3 or when we fee Fire by ftriking
the Eye, or fee Colours like the Eye of a Peacock's
Feather, by preffing our Eyes in either comes whilft
we look the other way. Where thefe and fuch like
caufes interpofe not, the Colour always anfwers to
the fort or forts of the rays whereof the Light coniifts,^
as I have conftantly found in what-ever Phsenomena ot
Colours 1 have hitherto been able to examin. I fhall in
the following Propofitions give inftances of this in the
Phenomena of chiefeft note.

PROP.

PROP. VIII. PROB. III.

B-j the difcove7'ed Trover ties of Light to explain the

Colours made hj Trijms, ^

Let ABC rcprefent a Prilm refrading the Light ofpj„-, i2,
the Sun, which comes into a dark Chamber through a
Hole F ? almoft as broad as the Prifm, and let M N
rcprefent a white Paper on which the refraded Light is
cart, and fuppofe the moft refrangible •t)r deepeft violet
making rays fall upon the fpace Ptt, the leaft refran-
gible or deepeft red-making rays upon the fpace T^,
the middle fort between the Indico-making aud blue-
making rays upon the fpace Q;^. , the middle fort of the
green-making rays upon the fpace R e , the middle fort
between the yellow-making and orange-making rays
upon the fpace ScT 7 and other intermediate forts upon
intermediate fpaces. For fo the fpaces upon which the
feveral forts adequately fall will by reafon of the diife-
rent rcfrangibility of thofe forts be one lower than ano-
\ther. Now if the Paper MN be fo near the Prifm that the
'fpaces P T and ttT do not interfere with one another, the
diftance between them T TT will be illuminated by all
the forts of rays in that proportion to one another which
they have at their very firft coming out of the Prilin,
and confequently be white. But the fpaces PT and ^
on either hand, will not be illuminated by them all,
and therefore will appear coloured. And particularly
at P, where the outmoft violet-making rays fall alone,
the Colour muft be the deepeft violet. At Q where the
violet-making and indico-making rays are mixed , it

(1 muft

[122]

miift be a violet inclining much to indico. At R where
the violet'making , indico-making , blue'making, and
one half of the green-making rays are mixed, their Co-
lours muft ( by the conftrudtion of the fecond Problem)
compound a middle Colour between indico and blue.
At S where all the rays are mixed except the red-ma-
king and orange-making,their Colours ought by the lame
Rule to compound a faint blue, verging more to green
than indie. And in the progrefs from S to T, tliis blue
will grow more and more faint and dilute, till at T,
where all the Colours begin to be mixed , it end in
whitenefs.

.^---So again, on the other fide of the w^hite at T, where
the leaft refrangible or utmoft red-making i-ays are alone
the Colour mult be the deepert red. At a the mixture
of red and orange will compound a red inclining to
orange. At e the mixture of red, orange, yellow, and
one half of the green mull compound a middle Colour
between orange and yellow. At x the mixture of all
Colours but violet and indico will compound a faint
yellow, verging more to green than to orange. And
this yellow will grow more faint and dilute continually/'
in its progrels from -^ to tt, where by a mixture of al^'
forts of rays it will become white.

Theie Colours ought to appear were the Sun's Light
perfedly white: But becaufe it inclines to yellow,theex-
cefs of the yellow-making rays whereby 'tis tinged with
that Colour, being mixed with the faint blue between
S and T, will draw it to a fa-int green. And fo the
Colours in order from P to T ought to be violet, indico,
blue, very faint green, white, faint yellow, orange, red.
Thus it is by the computation : And they that pleale to

view

[123]

/lew the Colours made by a Prifin will find it fo In
NJature.

Theie are the Colours on both fides the white when
he Paper is held between the Prifm, and the point X
vhere the Colours meet, and the interjacent white va-
liflies. For if the Paper be held ftill farther off from the
Mfm, the moft refrangible and leaft refrangible rays
vill be wanting in the middle of the Light, and the reft
)f the rays which are found there, will by mixture pro-
luce a fuller green than before. Alfo the yellow and
)lue will now become lefs compounded, and by con-
equence more intenfe than before. And this alfo
[grees with experience.

And if one look through a Prifm upon a white Objed
'ncompafled with blacknefs or darknefs, the reafon of
he Colours arifing on the edges is much the fame, as
vill appear to one that ihall a little confider it. If a
)lackObjed be encompaffed with a white one, the Co-
ours which appear through the Prifm are to be derived
rem the Light of the white one, fpreading into the Re-
;ionsof the black, and therefore they appear in a con-
,rary order to that, in which they appear when a white
)bjed: is furrounded with black. And the fame is to
le underftood when an Objed is viewed, whofe parts
re fome of them lefs luminous than others. For in the
borders of the more and lefs luminous parts, Colours
mght always by the lame Principles to arife from the
xcefs of the Light of the more luminous, and to be of
he fame kind as if the darker parts were black, but yet
0 be more faint and dilute.

Q ^ What

[124]

What is faid of Colours made by Prlfms may be eafil
applied to Colours made by the Glafles of Telefeop
or Microfcopes, or by the humours of the Eye. For i
the Objecl'glafs of a Telefcope be thicker on one fid
than on the other, or if one half of the Glafs, or on
half of the Pupil of the Eye be covered with any opak
fubftance : the Objed-glafs, or that part of it or of th
Eye which is not covered, may be conlidered as a Wedg
with crooked lides, and every Wedge of Glafs, orothe
pellucid fubftaQce, has the effed of a Prifm in refradin;
the Light which pafles through it.

How the Colours in the 9th and loth Experiment
of the firft Part arife from the different reflexibility 0
Light,is evident by what was there faid. But it is obfer
vable in the 9th Experiment, that whilft the Sun's di
red Light is yellow, the excefs of the blue-makin|
rays in the refleded Beam of Light M N, fuffices onb
to bring that yellow to a pale white inclining to blue
and not to tinge it with a manifeftly blue Colour. T(
obtain therefore a better blue, 1 ufed in Head of the yel
low Light of the Sun the white Light of the Clouds, bj.
varying a little the Experiment as follows.

EXPER. XVL

F^""". 1 5. Let H F G reprefent a Prifm in the open Air, and [
the Eye of the Spedator, viewing the Clouds by thei
Light coming into the Prifm at the plane tide FIGK
and reiieded in it by its bafe H E I G, and thence goin^
out through its plain fide H E F K to the Eye. Am
when the Prifm and Eye are conveniently placed, f(
that the Angles of incidence and reflexion at the baft

may be about 40 degrees, the Speftator will fee a Bow
M N of a blue Colour, running from one end of the
bafe to the other, with the concave fide towards him,,
and the part of the bafe IMNG beyond this Bow will
be brighter than the other part E M N H on the other
iide ol^ it. This blue Colour MN being made by no-
thing elfe than by reflexion of a fpecular fuperticies,
leeins fo odd a Phaenomcnon, and fo unaccountable for
by the vulgar Hypothelis of Fhilofophers, that I could-
not but think it deferved to be taken notice of. Now
for underftanding the realbn of it, fuppole the plane
ABC to cut the plane fides and bafe of the Prifm per-
pendicularly. From the Eye to the line BC, wherein that
plane cuts the bafe, draw the lines Sp and S t, in the
Angles Spc 50 degr. ;» andStc49 degr.-[s, and the
point / will be the limit beyond which none of the mofb
refrangible rays can pais through the bafe of the Prifm,
and be refracted, whole incidence is fuch that they may
be rctieded to the Eye ; and the point t will be the like
limit for the leaft refrangible rays, that is, beyond
which none of them can pafs through the bafe, whofe
incidence is fuch that by reflexion they may come to the
Eye. And the point r taken in the middle way between
p and t, will be the like limit for the meanly refrangible'
rays. And therefore all the refrangible rays which fall
upon the bafe beyond t, that is, between t and B, and
can comiC from thence to the Eye will be refleded thi-
ther : But on this fide t, that is, between t and c, many .
of thefe rays will be tranfmitted through the bafe.
And all the moft refrangible rays which fall upon the
bafe beyond p, that is , between p and B, and can by
reflexion come from thence to the Eye, will be reflected •

thithcr<» -

thither, but every where between t and c, many of
thefe rays will get through the bafe and be refraded ;
and the faine is to be underftood of the meanly refran-
gible rays on either fide of the point r. Whence it fol-
lows, that the bale of the Prifm muft: every where be-
tween t and B, by a total reflexion of all forts of rays to
the Eye, look white and bright. And every where
between p and C, by realbn of the tranfmiffion of many
rays of every fort, look more pale, obfcure and dark.
But at r, and in other places between p and t, where
all the more refrangible raj's are refleifled to the Eye,
and many of the lefs refrangible are tranfmitted, the
excefs of the moft refrangible in the reflected Light, will
tinge that Light with their Colour, which is violet and
blue. And this happens by taking the line Cp r t B any
where between the ends of the Prifm H G and E L

PROP. IX. PROB. IV.

jBy the difcovered Tro^erties of Light to explain the
Colours oj the Rjitn'h'vi>.

This Bow never appears but where it Rains in the
Sun-(hine, and may be made artificially by fpouting up
Water which may break aloft, and fcatter into Drops,
and fall down like Rain. For the Sun fliining upon thefe
Drops certainly caufes the Bow to appear to a Spefta^
tor itanding in a due pofition to the Rain and Sun. And
hence it is now agreed upon, that this Bow is made by
refradfion of the Sun's Light in Drops of tailing Rain.
This was underfi:ood by fome of the Ancients, and of
late more fully difcovered and explained by the Famous

y^ntonius

C 127 ]

uiintomm de'Dominis Archbifhop of 6')'i/^if(?, in his Book
^e Radtk Vt[m if7 Lucis^ publiflied by his Friend Bar*
tolm at Venice^ in the Year 1 6 1 1 , and written above
twenty Years before. For he teaches there how the
interior Bow is made in round Drops of Rain by two
refradions of the Sun's Light, and one reflexion be-
tween them, and the exterior by two refradions and
two forts of reflexions between them in each Drop of
Water, and proves his Explications by Experiments
made with a Phial full ofWater,and with Globes ofGlafs
filled with Water, and placed in the Sun to make the
Colours of the two Bows appear in them. The fame
Explication 'Des-Cartes hath purfued in his Meteors^
and mended that of the exterior Bow. But whilft they
underfliood not the true origin of Colours, it's neceffary
to purfue it here a little further. For underfl:anding
therefore how the Bow is made, let a Drop of Rain or
any other fpherical tranfparent Body be reprefented by
the Sphere BN FG, defcribed with the Center C, and Fig. 14 >
Semi-diameter CN. And let AN be one of the Sun's
rays incident upon it at N, and thence refraded to F,
where let it either go out of the Sphere by refradion to-
wards V, or be reflected to G ; and at G let it either go
out by refraction to R, or be reflected to H ; and at H
let it go out by refraction towards S, cutting the inci-
dent ray in Y ; produce A N and R G, till they meet in.
X, and upon A X and N F let fall the perpendiculars
CD and CE, and produce CD till it fall upon the cir-
cumference at L. Parallel to the incident ray A N draw
the Diameter B Q, and let the fine of incidence out of
Air into Water be to the line of refradion as I to
R. Now if you fuppofe the point of incidence N to

move

move from the point B, continually till it come to L,
t4ie Arch QF will firft increale and then decreafe, and
lb will the Angle AXR which the rays AN and GR
contain; and the Arch QF and Angle AXR w^ill be
biggell when ND is to CN as //hIrr to /^^ RR^
in which cafe N E will be to N D as a R to I. Alfo the
Angle AYS which the rays A N and HS contain will
firll decreafe, and then increafe and grow leaft when
ND is to CNas //fTRR to//8 RR, in which 4:afe
N E will be to N D as 5 R to I. And fo the Angle which
the next emergent ray ( that is, the emergent ray after
three reflexions ) contains with the incident ray AN
will come to its limit when ND is to CN as // ii-rr to
/^ 1 5 R R, in which cafe N E will be to N D as 4 R to I,
and the Angle which the ray next after that emergent,
that is, the ray emergent after four reflexions, con-
tains with the incident will come to its limit, when
N D is to C N/ as / ii-rr to //14 R R , in which cafe
N E will be to N D as 5 R to 1 ; and fo on infinitely,
the numbers 5, 8, 1 5, 24, ]5>c-. being gathered by conti-
nualaddition of the terms of the arithmetical progreflion
5,5,7, 9,isV. The truth of all this Mathematicians will
eaflly examine.

Now it is to be obferved, that as when the Sun comes
to his Tropicks, days increafe and decreafe but a very
little for a great while together ; fo when by increaf ng
the difl:ance C D, thefe Angles come to their limits,
they vary their quantity but very little for fome time
together, and therefore a far greater number of the rays
which tall upon all the points N in the Quadrant
BL, fhall emerge in the limits of thefe Angles,
^then in any other inclinations. And further it is

[129]

to be obierved, that the rays which differ in refrangl-
biHty will have different limits of their Angles of emer-
gence, and by confequence according to their different
degrees of refrangibility emerge moll copioufly in dif-
ferent Angles, and being feparated from one another
appear each in their proper Colours. And what thofe
Angles are may be eafily gathered from the foregoing
Theorem by computation.

For in the leaft refrangible rays the fines I and R (as
was found above) are 108 and 81, and thence by
computation the greateft Angle AXR will be found
42 degrees and 0. minutes, and the leaft Angle AYS,
50 degr. and 57 minutes. And in the moft refrangible
rays the fines I andR are 109 and 81, and thence by
computation the greateft Angle AXR will be found
40 degrees and 1 7 minutes, and the leafi: Angle AYS
54. degrees and 7 minutes.

Suppofe now that O is the Spectator's Eye, and OP a line fig. 1 5 ,
drawn parallel to the Sun's rays, and let PO E, POP,
POG, POH, be Angles of 40 degr. i7min. 41 degr.
2 min. 50 degr. 57 min. and 54 degr. 7 min. refpedively,
and thefe Angles turned about their common fide O P,
Ihall with their other fides OE, OF; OG, OH de-
defcribe the verges of two Rain-bows AFBE and
CHDG. For if E, F, G, H, be Drops placed aiw
where in the conical fuperficies defcribed by O E, O F,
OG, OH, and be illuminated by the Sun's rays SE,
SF, SG, SH; the Angle SEO being equal to the
Angle POE or 40 degr. 17 min. fhall be the greateft
Angle in which the moft refrangible rays can after one
reflexion be refraded to the Eye, and therefore all the
Drops in the line O E fhall fend the moft refrangible

R I'j^ys

[130]

rays moft copiouny to the Eye, and thereby ftiike the
fenles with the dcepeft violet Colour in that region.
And in like manner the Angle SFO being equal to:
the Angle P OF, or 4:2 deg. 2 min. fhall be the greateft
in which the lead refrangible rays after one reflexion
can emerge out of the Drops, and therefore thofe rays
(hall come moft copioufly to the Eye from the Drops in
the line O F, and ftrike the fenles with the deepeft red
Colour in that region. And by the fame argument,
the rays which have intermediate degrees of xefrmigibi-
lity fhall come moft copioufly from Drops between
E and F, and ftrike the fenles with the intermediate
Colours in the order which their degrees of refrangibi-
Uty require , that is, in the progrefs from E to F, or
from the inflde of the Bow to the outiide in this order,
violet, indico,blue, green, yellow,orange, red. But the
violet, by the mixture of the white Light of the Clouds,
will appear faint and incline to purple.

Again, the Angle S G O being equal to Angle P O G,,
or 50 gr. 51 min. fhall be the leaft Angle in which the
l,eaft refrangible rays can after two reflexions emerge out
of the Drops,and therefore the leaft refrangible rays fliall
come moft copioufly to the Eye from the Drops in the
line O G, and ftrike the fenfe with the deepeft red in
that region. And the Angle S HO being equal to the
Angle P OH or 54. gr. 7 min. fliali be the leaft Angle in
which the moft refrangible rays after two reflediohs can
emerge out of the Drops, and therefore thofe rays fhall
come moft copioufly to the Eye from the Drops in the
line O H, and ftrike the fenles with the deepeft violet in
that region. And by the fame argument, the Drops in
the regions between G and H fhall ftrike the fenfe with

the

CI30

the intermediate Colours in the order which tlicir de-
grees of refrangibility require^ that is, in the progrefs
from G to H, or from the iniide of the Bow to the out-
lide in this order, red, orange, yellow, green, blue, in-
dico, violet. And fince thefe four lines O E, O F, O G.
O H, may be iituated any where in the above-mentioned
conical fuperficies, what is faid of the Drops and Co-
lours in thefe lines is to be underftood of the Drops
and Colours every where in thofe fuperficies.

Thus fliall there be made two Bows of Colours, an
interior and ftronger, by one reflexion in the Drops,
and an exterior and fainter by two ; for the Light be-
comes fi3 inter by every reflexion. And their Colours
{hall ly in a contrary order to one another, the red of
both Bows bordering upon the fpace G F which is be-
tween the Bows. The breadth of the interior Bow
EOF meafured crofs the Colours fnall be i degr. 45 min.
-ind the breadth of the exterior GOH ihall be -9
degr. lomin. and the difxance between them GOF
ihall be 8 gr. 5 5 min. the greatefl: Semi-diameter of the
innermoft, that is, the Angle POF being 4a gr. 2 min.
gnd the leaft Semi-diameter of the outermoil P O G, be-
ing 50 gr. 57 min. Thefe are the meafures of the Bows,
; as they would be w^re the Sun but a point ; for by the
breadth of his Body the breadth of the Bows will be in-
creafed and their diftance decreafed by half a deg,ree,
and io the breadth of the interior Iris wall be 1 degr.
15 min. that of the exterior 9 degr. 40 min. their di-
:iJance 8 degr. 25 min. the greateft Semi-diameter of the
interior Bow 42 degr. 17 min. and the leaft of the ex-
.terior 50 d^gr. 4a mJn. And fuch are the dimeniions
r.iof the Bows in the Heavens found to be very nearly,
■ R 2 when

[t3'2]

when tht'ii" Colours appear ftrong and pcift'ct. For
once, by fuch means as I then had, I nieafured the
greateft Semi-diameter of the interior Iris about 4.2 de-
grees, the breadth of the red, yellow and green in that
Iris 63 or 64. minutes, befides theoutnToft faint red ob-
fcured by brightncfs of the Clouds, for which we
may allow 3 or 4 minutes more. The breadth of the
blue was about 4.0 minutes more be (ides the violet,
which was fo much obfcured by the brightnefs of the
Clouds, that 1 could not meaiure its breadth. But
fuppofing the breadth of the blue and violet together
to equal that of the red, yellow and green together, the
whole breadth of this Iris will be about 1^ degrees as
above. The leaft diftance between this Iris and the ex-
terior Iris was about 8 degrees and 50 minutes. The ex-
terior Iris was broader than the interior, but fo faint,
efpecially on the blue tide, that I could not meaiure its
breadth diftindly. At another time when both Bows
appeared more diftinft, I meafured the breadtli of the
interior Iris 2 gr. ic, and the breadth of the red, yel-
low and green in the exterior Iris, was to the breadth
of the fame Colours in the interior as 5 to a.

This Explication of the Rain-bow is yet further con-
firmed by the known Experiment ( made by Antonim
de Dominis and T)es -Cartes) of hanging up any where
in the Sun-fhine a Gl ifs-Globe filled with Water, and
viewing it in fuch a pofture that the rays which come
from the Globe to the Eye may contain with the Sun's
rays an Angle of either 4a or 50 degrees. For if the
Angle be about 4.1 or 4.3 degrees, the Spectator ( fup-
pofe at O) (hall fee a full red Colour in that fide of the
Globe oppofed to the Sun as 'tis reprefented at F, and

[133.3

if that Angle become lets ( luppofe by depreffing tlie
Globe to E) there will appear other Coloms, yellow,
green and blue fucceffively in the lame (ide of the Globe.
But if the Angle be made about 50 degrees (luppole by
lifting up the Globe to G)there will appear a red Colour
in that fide of the Globe towards the Sun, and if the
Angle be made greater (fuppofe by lifting up the Globe
to H) the red will turn fucceffively to the other Colours
yellow, green and blue. The lame thing I have tried by
letting a Globe reft, and raihng or depreffing the Eye,
or otherwife moving it to make the Angle of a juft
magnitude.

1 have heard it reprefented, that if the Light of a
Candle be refraded by a Prifm to the Eye ; when the
blue Colour falls upon the Eye the Spedator fhall lee
red in the Prifm, and when the red fliUs upon the Eye
he fhall fee blue ; and if this were certain, the Colours
of the Globe and Rain-bow ought to appear in a con-
trary order to what we find. But the Colours of the
Candle being very faint, the miftake feems to arife from
the difficulty of difcerning what Colours fall on the
Eye. For, on the contrary, I have fometimes had oc-
calion to obferve in the Sun's Light refraded by a Prifm,
that the Spe(ftator always fees that Colour in the Prifm
which falls upon his Eye. And the fame I have found
true alfo in Candle-Light. For when the Prifm is mo-
ved (lowly from the line which is drawn directly from the
Candle to the Eye,the red appears firft in the Prifiii and
then the blue, and therefore each of them is feen when
it falls upon the Eye. For the red paffes over the Eye
firft, and then the blue.

The

1 134-1

The Light wiiich comes through Drops of Ruin hy
two refractions without any reflexion, ought to appear
Ihongeft at tlie diftancc of about a 6 degrees from the
Sun, and to decay gradually both ways as the dilbnce
from him increales and dccreafes. And the fame is to
be underftood of Light tranfmitted through fpherical
Hail-ftoncii. And if the Hail be a little hatted, as it
often is, the Light tranfmitted may grow fo ftrong at
a little lefs diftance than that of 16 degrees, as to form
a Halo about the Sun or Moon ; which Halo, as often
as the Hail-ftones are duly figured may be coloured,
and then it muft be red within by the lead refrangible
rays,and blue without by the moft refrangible ones,efpe-
xially if the Hail-flones have opake Globules of Snow in
their center to intercept the Light within the Halo ( as
Hugenim has obferved) and make the inlide thereof more
-diftindly defined than it would otherwife be. For
luch Hail-flones, though fpherical, by terminating the
Light by the Snow, may make a Halo red within and
colourlefs without, and darker in the red than with=
out, as Halos ufe to be. For of thofe rays which pafs
elofe by the Snow the rubriform will be leaft refradtedj
and fo come to the Eye in the dire6teft lines.

The Light which paiTes through a Drop of rain after
two refractions, and three or more reflexions, is fcarce
ftrong enough to caule a fenfible Bow ', but in thofe Cy-
linders of Ice by w^hich Hugemm explains ih^ Tiifheha^
it may perhaps be fenfibk-

PROP.

P R O p. X. P R O B. V.

Bj the dtfcovered properties of Light to explain the fer^
manent Colours of natural Bodies.

Thefe Colours arlfe from hence, that fome natural
Bodies refled fome forts of rays, others other forts more
copiouQy than the reft. Minium reflects the leaft re-
fi'angible or red-making rays raoft copioufly, and thence
appears red. Violets reflect the moft refrangible, moft
copioufly, and thence have their Colour, and fo of other
Bodies. Every Body reflects the rays of its own Colour
more copioufly than the reft, and from their excefs and
predominance in the reflected Light has its Colour.

EX PER. xvir.

For if the homogeneal Lights obtained by the folu*
tion of the Problem propofed in the 4.th Propolition of
the firft Book you place Bodies of feveral Colours, you
will find, as I have done, that every Body looks moft
fplendid and. luminous in the Light of its own Colour,
Cinnaber in the homogeneal red Light is moft refplcn-
dent, in the green Light it is manifeftly lefs refpleiv
dent, and in the blue Light ftill lefs. Indico in the
violet blue Light is moft refplendent, and its fplendoK
is gradually diminiflied as it is removed thence by de-
grees through the green and yellow Light to the red.
By a Leek the green Light, and next that the blue and
yellow which compound green, are more ftrongly re-

fleaed.

[13^]

fleeted than the other Colours red and violet,and fo of the
reft. But to make thcfe Experiments the more manifeft,
luch Bodies ought to be choten as have the fulleft and
moft vivid Colours, and twoof thole Bodies are to be
compared together. Thus, tor inftance, if Cinnaber
and ultra marine blue, or fome other full blue be
held together in the homogeneal Light, they will both
appear red, but the Cinnaber will appear of a ftrongly
luminous and refplendent red, and the ultra marine
blue of a faint obfcure and dark red ; and if they be
held together in the blue homogeneal Light they will
both appear blue, but the ultra marine will appear of
a ftrongly luminous and relplendent blue, and the
Cinnaber of a faint and dark blue. Which puts it out
of difpute , that the Cinnaber reflects the red Light
much more copiouily than the ultra marine doth, and
the 2iltra marine retlefts the blue Light much more co-
pioufly than the Cinnaber doth. The fame Experiment
may be tryed fucccsfully with red Lead and Indico, or
v,dth any other two coloured Bodies, if due allowance
be made for the different ftrength or weaknefs of their
Colour and Light.

And as the reafon of the Colours of natural Bodies is
evident by thefe Experimenrs, fo it is further confirmed
and put paft difpute by the two firft Experiments of the
firft Book, whereby 'twas proved in fuch Bodies that
the reileded Light which differ in Colours do differ alfo
hi degrees of refrangibility. For thence it's certain,
that fome Bodies retie*^ the more refrangible, others
the lefs refrangible rays more copioufly.

And

[137]

And that this is not only a true reafon of thefe Co-
lours, but even the only realbn may appear further
from this conlideration, that the Colour of homogeneal
Light cannot be changed by the reflexion of natural
Bodies.

For if Bodies by reflexion cannot in the leaft change
the Colour of any one fort of rays, they cannot appear
coloured by any other means than by refleding thofe
which either are of their own Colour, or which by
mixture muft produce it.

But in trying Experiments of this kind care mufl: be
had that the Light be fufficiently homogeneal. For if
Bodies be illuminated by the ordinary prifmatick Co-
lours, they will appear neither of their ow^n day-light
Colours, nor of the Colour of the Light cafl: on them,
but of fome middle Colour between both, as I have
found by Experience. Thus red Lead ( for inftance )
illuminated with the ordinary prifmatick green will
not appear either red or green, but orange or yellow,
or between yellow and green accordingly, as the green
Light by which 'tis illuminated is more or lefs com-
pounded. For becaufe red Lead appears red when il-
luminated with white Light, wherein all forts of rays
are equally mixed, and in the green Light allforts of
rays are not equally mixed, the excefs of the yellow-
making, green-making and blue-making rays in the
• incident green Light, will caufe thofe rays to abound
fo m.uch in the reflected Light as to draw the Colour
from red towards their Colour. And becaufe the red
Lead reflects the red-making rays moft copioufly in
proportion to their number, and next after them the
orange-making and yellow-making rays ; thefe rays in

S the

the refleded Light will be more in proportion to tlie
Light than they were in the incident green Light, and
thereby will draw the refleded Light from green to-
wards their Colour. And therefore the red Lead will ap-
pear neither red nor green,butofaColour between both.
In tranfparently coloured Liquors 'tis oblcrvable,
that their Colour ufes to vary with their thicknefs.
Thus, for inftance, a red Liquor in a donical Glafs
held between the Light and the Eye, looks of a pale
and dilute yellow at the bottom wiiere 'tis thin, and a
little higher where 'tis thicker grows orange,and where
'tis ftill thicker becomes red, and where 'tis thickeft
the red is deepeft and darkeft . For it is to be conceived
that fuch a Liquor ft ops the indico-making and violet-
making rays moft ealily, the blue- making rays more
difficultly, the green-making rays ftill more difficultly,
and the red-making moft difficultly : And that if the
thicknefs of the Liquor be only fo much as fuffices to
ftop a competent number of the violet-making and in-
dico-making rays, without diminilhing much the num-
ber of the reft, the reft muft (by Prop. 6. Ltl^. 'i. j com-
pound a pale yellow^ But if the Liquor be fo much
thicker as to ftop alfo a great number of the blue-making
rays, and fome of the green-making, the reft muft com-
pound an orange ; and w^here it is fo thick as to ftop
alfo a great number of the green-making and a confi-
derable number of the yellow-making, the reft muft
begin to compound a red, and this red muft grow deeper
and darker as the yellow making and orange-making
rays are more and more ftopt by increaftng the thick-
nefs of the Liquor, lb that few rays beiides the red-
making can get through^

[ 1 39 ]

Of this kind is an Experiment liitely related to me by
Mr. HaUe>j^ who, in diving deep into the Sea, found
in a clear Sun-fhine day, that when he was lunk many
Fathoms deep into the Water, the upper part of his
Hand in which the Sun Ihone dire6tly through the
Water looked of a red Colour, and the under part of
his Hand illuminated by Light retieded from the Water
below looked green. For thence it may be gathered,
that the Sea .water refleds back the violet and blue-
making rays moll: eaiily, and lets the red-making rays
pals moft freely and copioully to great depths. For
thereby the Sun's dired Light at all great depths, by
reafon of the predominating red-making rays, mult
appear red ; and the greater the depth is, the fuller
and intenfer muft that red be. And at fuch depths as
the violet-making rays fcarce penetrate unto, the blue-
making, green-making and yellow-making rays being
relieved from below more copioully than the red-making
ones, mull compound a green.

Now if there be two Liquors of full Colours, fup-
pofe a red and a blue, and both of them fo thick as
fuffices to make their Colours fufficiently full ; though
either Liquor be fufficiently tranfparent apart, yet
will you not be able to fee through both together. For
if only the red-making rays pals through one Liquor,
and only the blue-making through the other, no rays
can pafs through both. This Mr. Hook tried cafually
with Glafs-wedges filled with red and blue Liquors,
and was furprized at the unexpected event, the reafon
of it being then unknown 3 which makes me truft the
more to his Experiment, though 1 have not tryed it
my felf. But he that would repeat it, muft take care
tiie Liquors be of very good and full Colours.

S 2 Now

[140]

Now whilfl: Bodies become coloured by reliedling or
tranfmitting this or that fort of rays more copiouOy than
the reft, it is to be conceived that they ftop and ftifle in
themfelves the rays which they do not retiector tranfmit.
For if Gold be foliated and held between your Eye and
the Light, the Light looks blue, and therefore maffy Gold
lets into its Body the blue.making rays to be retleded
to and fro within it till they be ftopt and ftifled, whilft
it retlecls the yellow-nwking outwards, and thereby
looks yellow. And much aftei' the fame manner that
Leaf-gold is yellow by reflected, and blue by tranfmit-
ted Light, and mafly Gold is yellow in all portions of
the Eye ; there are fome Liquors as the tindure of
Lignum iSfefhrit'icum^ and fome forts ot Glafs which
tranfmit one fort of Light moft copioufly, and reileft
another fort, and thereby look of feveral Colours, ac-
cording to the polition of the Eye to the Light. But if
thefe Liquors or Glaffes were ^io thick and maffy that
no Light could get through them, 1 queftion not but
that they would like all other opake Bodies appear of
one and the fame Colour in all pofitions of the Eye,
though this 1 cannot yet affirm by experience. For all
coloured Bodies, fo tar as my Obfervation reaches, may
be leen through if made fufficiently thin,., and therefore
are in fome meaiure tranfparent, and differ only in de-
grees of tranfparency from tinged tranfparent Liquors •_
thefe Liquors, as well as thole Bodies, by a fufficient
thicknefs becoming opake. A tranfparent Body which
looks of any Colour by tranfmJtted Light, may alio
look of the fame Colour by reflefted Light, the Light
of that Colour bein^ rerleded bv the furtlier furtace of
the Body, or by the Air beyond it. And then the re-
tleded Colour will be diminilhe'Uand perhaps ceale, by

jnaking

[ 141 ]

making the Body very thick, and pitching it on the
hick-fide to diminifli the reflexion of its further furface,
fo that the Light reiiefted from the tinging particles
.may predominate. In fuch cafes, the Colour of the re-
fiedled Light will be apt to vary from that of the Light
tranfmitted. But whence it is that tinged Bodies and
Liquors refled: fome fort of rays, and intromit or trans-
mit other forts, ihall be laid in the next Book. In this
Propofition 1 content my felf to have put it paft difpute,,
that Bodies have fuch Properties, and thence appear
coloured.

PROP. XL PROB. VL

B'i mixing coloured Lights to corn-pound a Bcarri of Ltg/jp
of the jame Colour and Mature isoith a Beam of the Suns-
direSi JL'rght^ and therein to experience the truth of the.
foregoing Tro-^o fit tons.

Let A B Cab c reprefent a Prifm by which the Sun's Fig- i^»
Light let into a dark Chamber through the Hole F, may
be refraded towards the Lens M N, and paint upon it
at p, q, r, s and t, the ufual Colours violet, blue, green^
yellow and red, and let the diverging rays by the re-
iradion of this Lens converge again towards X, and
thcrc,by the mixture of all thofe their Colours,compound
a white according to what was fhewn above. Then let
another Prifm DEGdeg, parallel to the former, be
placed at X, to refrad that white Light upwards to-^
wards Y. Let the refrading Angles of the Prifms^
and their dillances from the Lens be equal, fo that the
rays which converged from the Lens towards X, and".
without refradion, would there have croffed and diver-
ged again, may by the refraction of the fecondPriim be.

reduced.

reduced into Parallelifm and divcrsie no more. For

then thole rays will recompofe a Beam of white Light

XY. If the refrafting Angle of cither Prilm be the

bigger, that Prifm mull be lb much the nearer to the

Lens. You will know when the Prifms and the Lens

are well let together by obferving if the Beam of Light

XY which comes out of the fecond Prifm be perfedly

white to the very edges of the Light, and at all diftan-

cesfrom the Prifm continue perfectly and totally white

like a Beam of the Sun's Light. For till this happens,

the polition of the Prifms and Lens to one another mull

be corrcdted, and then if by the help of a long Beam of

Wood, as is reprefented in the Figure, or by a Tube,

or fome other fuch inftrument made for that purpofe,

they be made faft in that lituation, you may try all the

lame Experiments m this compounded Beam of Light'

XY, which in the foregoing Experiments have been

made in the Sun's direct Light. For this compounded

Beam of Light has the lame appearance, and is endowed

with all the fame Properties with a diredt Beam of the

Sun's Light, lb far as my Obfervation reaches. And in

trying Experiments in this Beam you may by ftopping

any of the Colours p, q, r, s and t, at the Lens, fee how

the Colours produced in the Experijnents are no other

than thofe which the rays had at the Lens before they

entered the compoiition of this Beam : And by confe-

cjuence that they arife not from any new modifications

of the Light by refractions and reflexions, but from the

various feparations and mixtures of the rays originally

endowed with their colour-making qualities.

So, for inrtance, having with a Lens 4.; Inches broad,
and two Prifms on either Hand 6^ Feet diftant from the
Lens, made fuch a Beam of compounded Light : to

examin

[ H3 ]

txamin the reaibn of the Colours made by Prifms, I
refraded this compounded Beam of Light XY with
another Prifm H I K k h, and thereby caft the ufual pi if-
matick Colours PQRST upon the iPaper LV placed be-
hind. And then by Hopping any of the Colours p, q,..
r, s, t, at the Lens, 1 found that the fame Colour would
vanifli at the Paper. So if the purple P was flopped at
the Lens, the purple P upon the Paper would vanifh,
and the reft of the Colours would remain unaltered,
unlets perhaps the blue, fo far as fome purple latent ni
it at the Lens might be feparated from it by the fol-
lowing refractions. And lb by intercepting the green
upon the Lens, the green R upon the Paper would va-
nifh, and fo of the reft ; which plainly (hews, that as
the white Beam of Light X Y was compounded of fe-
ve Lights varioufly coloured at the Lens, lb the Co-
lours which afterwards emerge out of it by new refra-
ftions are no other than thofe of which its whitenefs
was compounded. The refraction of the Prifm H I K
kh generates the Colours PQRST upon the Paper,
not by changing the colorific qualities of the rays, but
by feparating the rays which had the very fame colorific
qualities before they entered the compofition of the re-
fraded Beam white of Light X Y. For otherwife the rays
which were of one Colour at the Lens might be of ano-
ther upon the Paper, contrary to what we find.

So again, to examin the reafon of the Colours of na-
tural Bodies, I placed fuch Bodies in the Beam of Light
XY, and found that they all appeared there of thofe
their own Colours which they have in Day-light, and
that thofe Colours depend upon the rays which had the
fame Colours at the Lens before they entred the compo-

lition

lition of that Beam. Thus, for inftance,Cinnaber illumi-
nated by this Beam appears of the fame red Colour a-- in
Day-light , and if at the Lens you intercept the green-
making and blue-making rays, its rednefs will become
more full and lively : But if you there intercept the red-
making rays, it will not any longer appear red, but be-
come yellow or green, or of Ibme other Colour, accor-
ding to the forts of rays which you do not intercept.
So Gokl in ti\is Light XY appears of the lame yellow
Colour as in Day-light, but by intercepting at the Lens a
due quantity of the yellow-m^aking rays it will appear
white like Silver (as 1 have tryed) which fhews that its
yellownels ariles from the excefs of the intercepted rays
tinging that whitenels with their Colour when they are
let pafs. ^ot\\Q m(\x(\on oi LigniimNefhrittcum (as I
have alio tryed ) when held in this Beam of Light X Y,
looks blue by the refieded part of the Light, and yellow
by the tranfmitted part of it, as when 'tis viewed in Day-
light, but if you intercept the blue at the Lens the infu-
fion will lofe its reflected blue Colour, whilit its tranf-
mitted red remains perfed: and by the lofs of ibme blue-
making rays wherewith it was allayed becomes morein-
tenfe and full. And, on the contrary, if the red and orange-
making rays be intercepted at Lens, the infufion will
loie its tranfmitted red, whilfl its blue will remain and
become more full and perfect. Which fhews, that the in- '
fuiion does not tinge the rays with blue and yellow, but
only tranlmit thofe moft copiouily which were red-ma-
king before, and reflefts thole moll copioufly which were
blue-making before. And after the lame manner may the
peafons of other Phaenomena be examined, by trying
them in this artificial Beam of Light X Y.

THE

Book I. Part I. Plate 1.

Book I. Part n. Plate E.

X Book! Part H. Plate m.

F£^.l^

Bookl. I'aitl.i'iatp

THE ''"^

SECOND BOOK

O F

O P T I C K S.

PART I.

O^fervations concerning the Reflexions^ Refradiions^ and
Colours of thin tranjfarent Bodies,

IT has been obferved by others that tranfparent
Subftances, as Glafs, Water, Air, he. when made
very thin by being blown into Bubbles, or otherwife
formed into Plates, do exhibit various Colours accor-
ding to their various thinnefs, although at a greater
thicknefs they appear very clear and colourleis. In
the former Book I forbore to treat of thefe Colours,
becaufe they feemed of a more difficult coniideration,
and were not neceffary for eftablilhing the Properties
of Light there difcourfed of. But becaufe they may
conduce to further difcoveries for completing the
Theory of Light, efpecially as to the conftitution of
the parts of natural Bodies, on which their Colours or
Tranfparency depend ; I have here let down an ac-
count of them. To render this Difcourfe Ihort and
diftind, I have lirft defcribed the piincipal of my

A a Obfer=

[2]

Obfervations, and then confidered and made ufe of
them. The Oblervations are thele.

O B S. I.

Compreffing two Prifms hard together that their
Sides (which by chance were a very httle convex)might
fomewhere touch one another : 1 found the place in
which they touched to become abibUitely traniparent,
as if they had there been one continued piece of Glafs.
For when the Light fell fo obliquely on the Air, which
in other places was between them,as to be all relieved ;
it feemed in that place of contad: to be wholly tranf-
mitted, infomuch that when looked upon, it appeared
like a black or dark Spot, by reafon that little or no
fenfible Light was reflected from thence, as from other
places; and when looked through it feemed (as it were)
a hole in that Air which was formed into a thin Plate,
by being compreffed between the Glaffes. And through
this hole Objects that were beyond might be Ci^tn di-
ftindly, which could not at all be feen through other
parts of the Glafles where the Air was interjacent. Al-
though the Glafles were a little convex, yet this tranf-
parentSpot was of a coniiderable breadth,which breadth,
feemed principally to proceed from the yielding inwards
of the parts of the GlalTes, by reafon of their mutual
preffure. For by preffing them very hard together it
would become much broader than otherwife.

OBS.

[3]

O B S. II.

When the Plate of Au', by turnuig thePiifms about
their common Axis, became lb little inclined to the in-
cident Rays, that Ibme of them began to be tranfmit-
ted, there arofe in it many flender Arcs of Colours
which at firft were ihaped almoft like the Conchoid,
as you fee them delineated in the firft Figure. Andi^^ig- ^
by continuing the motion of the Prifms, thefe Arcs in-
creafed and bended more and more about the faid tranf-
parent Spot, till tliey were completed into Circles or
Rings incompaffing it, and afterwards continually grew
more and more contracted.

Thefe Arcs at their firft appearance were of a violet
and blue Colour, and between them were white Arcs
of Circles, which prefently by continuing the motion of
the Prifms became a little tinged in their inward Limbs
with red and yellow, and to their outward Limbs the
blue was adjacent. So that the order of thefe Colours
from the central dark Spot, was at that time white,
blue, violet j black ; red, orange, yellow, white, blue,
violet, 't^c. But the yellow and red were much fainter
than the blue and violet.

The motion of the Prifms about their Axis being con-
tinued, thefe Colours contracted more and more,ilirink-
ing towards the whitenefs on either fide of it, until they
totally vanifhed into it. And then the Circles in thofe
parts appeared black and white, without any other Co-
lours intermixed. But by further moving the Prifms
about, the Colours again emerged out of the whitenefs,
the violet and blue as its inward Limb, and at its out-

A a 2 ward

[4]

ward Limb the red and yellow. So that now their order
from the central Spot was white, yellow, red ; black ;
violet, blue, white, yellow, red, oc. contrary to what
it was before.

O B S. III.

When the Rings or fome parts of them appeared only
black and white, they were very diftind: and well de-
fined, and the backnefs fcemed as intenfe as that of
the central Spot. Alio in the borders of the Rings,
where the Colours began to emerge out of the white-
nefs, they were pretty diftintt, which made them vi-
fible to a very great Multitude. I have fometimes
numbred above thirty Sncceffions ( reckoning every
black and white Ring for one Succeffion ) and feen
more of them^ which by reafon of their fmalnefs I could
not number. But in other Pofitions of the Prifms, at
which the Rings appeared of many Colours, I could not
diftinguifh above eight or nine of them, and the exte^
rior of thofe were very confuted and dilute.

In thefe two Obfervations to fee the Rin^s diftinft.
and without an-y oth^r Colour than black and white,!
found it neceflary to hold my Eye at a good diftance
from them. For by approaching nearer, although in the
fame inclination of my Eye to the plane of the Rings,
there emerged a blueifh Colour out of the white,
which by dilating it felf more and more into the black
rendred the Circles lefs diftintl:, and left the white a
little tinged with red and yellow. I found alio • by
looking through a flit or oblong hole , which was
narrower than the Pupil of my Eye, and held clofe t€^

it.

[5]

it parallel to the Prifms, 1 could fee the Circles much
dilHnder and vifible to a far greater number than
Gtherwife,

O B S. IV.

To obferve more nicely by the order of the Colours
which arofe out of the white Circles as the Rays be-
came iefs and Icfs inclined to the plate of Air; 1 took
two Objed GlafTes, the one a Plano-convex for a four-
teen-foot Telefcope, and the other a large double con-
vex for one of about fifty-foot; and upon this,laying the
other with its its plane-fide downwards, I prefled them
llowly togethcr^to make the Colours fucceflively emerge
in the middle of the Circles,, and then flowly lifted
the upper Glafs from the lower to make them fuccef-
fively vanifli again in the fame place. The Colour,,
which by preffing the Glaffes together emerged lafi: in
the middle of the other Colours, would upon it& firft
appearance look like a Circle of a Colour almoft uni-
form from the circumference to the center , and by
compreffing the Glaffes ftill moi"e, grow continually
broader until a new Colour emerged in its center, and
thereby it became a Ring encompafling that new Co-
lour. And by comprefling the Glafles ftill more, the
Diameter of this Ring would encreafe, and the breadth
of its Orbit or Perimeter decreafe until another new
Colour emerged in the center of the laft : And fo on
iintil a third, a fourth, a fifth, and other following
new Colours fucceilively emerged there^ and became
Rings encompaffing the innermoft Colour, the laft of
which was the black Spot. And, on the contrary, by

lifting

lifting up the upper Glals trom the lower, the diameter
of the Rings would decreafe, and the breadth of their
Orbit encreafe, until their Colours reached fucceflively
to the center ; and then they being of a conliderable
breadth, I could more ealily difcern and dirtinguifli
their Species tlian before. And by this means 1 ob-
ferved their Succeffion and Quantity to be as fol-
loweth. .^ .

Next, to the pellucid central Spot made by the con-
tad of the GlafTcs lucceeded blue, white, yellow, and
red, the blue was fo little in quantity that I could not
difcern it in the circles made by the Prifms, nor could
I well diftinguifli any violet in it, but the yellow and
red were pretty copious, and ieemed about as much
in extent as the white , and four or five times more
than the blue. The next Circuit in order of Colours
immediately encompaffing thefe were violet, blue,
green, yellow, and red, and thefe were all of them co-
pious and vivid, excepting the green, which was very
little in quantity, and Ieemed much more faint and
dilute than the other Colours. Of the other four, the
violet was the leaft in extent , and the blue lefs than
the yellow or red. The third Circuit or Order was
purple, blue, green, yellow, and red ; in which the
purple ieemed more reddifh than the violet in the
former Circuit, and the green was much more confpi-
cuous, being as brifque and copious as any of the other
Colours, except the yellow ; but the red began to be
a little faded, inclining very much to purple. After
this fucceeded the fourth Circuit of green and red. The
green was very copious and lively, inclining on the one
lide to blue, and on the other fide to yellow. But in

this

[7J .

this fourth Circuit there was neither violet, blue, nor
yellow, and the red was very imperfecl: and dirty.
Alfo the fucceeding Colours became more and more im-
perfed and dilute, till after three or four Revolutions
tiiey ended in perfect whitenefs. Their Form, when the
GliiHes weremoft comprcfTed fo as to make the black
Spot appear in the Center, is delineated in the Second
Figure ', where «, <^, r, ^, e ; f, g^ /j, z, h. : /, w, w, o, ^ ; q^ r : Fig. 2.
J-, t : Vyx:y denote the Colours reck'ned in order from _
the center, black, blue, white, yellow, red : violet,
blue, green, yellow, red : purple, blue, green, yellow,
red : green, red : greenilli blue, red : greeniih blue,
pale red : greeniili blue, reddiih white.

O B S. V.

To determine the interval of the GlafTes, or thick-
nefs of the interjacent Air, by which each Colour was
produced, I meafured the Diameters of the firft fix
Rings at the moft lucid part of their Orbits, and fqua-
ring them, I found their Squares to be in the Arith-
metical Progreffion of the odd Numbers, i . 5. 5. 7. 9. 1 1 .
And fince one of thefe Glaffes was Plain, and the other
Spherical, their Intervals at thofe Rings muft be in the
fame Progreffion. I meafured alfo the Diameters of
the dark or faint Rings between the more lucid Co-
lours, and found their Squares to be in the Arithme-
tical Progreffion of the even Numbers, a. 4.. 6. 8. 10. la*
And it being very nice and difficult to take thefe mea-
fures exadly ; 1 repeated them at divers times at divers
partsof the Glaffes, that by their Agreement I might
be confirmed in them. And the fame Method I ufed in

deter-

determining fome others of t^e following Obferva-
tions.

O B S. VI.

The Diameter of the fixth Ring at the moft lucid
part of its Orbit was £, parts of an Inch, and the Dia-
meter of the Sphere on which the double convex Ob-
jedli-Glafs was ground was about loa Feet, and hence
I gathered the thicknefs of the Air or Aereal Interval
of the Glaffes at that Ring. But fome time after, fuf-
peding that in making this Obfervation I had not de-
termined the Diameter of the Sphere with fufficient ac-
curatenefs, and being uncertain whether the Plano-
convex Glafs was truly plain, and not fomething con-
cave or convex on that lide which I accounted plain ;
and whether 1 had not prefled the Glaffes together, as

I often did, to make them touch (for by preffmg fuch
Glaffes together their parts eafily yield inwards, and
the Rings thereby become feniibly broader , than they
would be, did the Glaffes keep their Figures) I re-
peated the Experiment, and tound the Diameter of
the iixth lucid Ring about 7;^ parts of an Inch. I re-
peated the Experiment alfo with fuch an Objed-Glafs
of another Telefcope as I had at hand. This was a double
convex ground on both fides to one and the fame
Sphere, and its Focus was diftant from it 8^j Inches.
And thence, if the Sines of incidence and refra6tion of
the bright yellow Light be affumed in proportion as

II to 17, the Diameter of the Sphere to which the
Glafs was figured will by computation be found 1 82 In-
dies. This Glafs 1 laid upon a flat one, fo that the

black

[9]

black Spot appeared in the middle of the Rings of Colours
without any other prelTure than that of the weight of
the Glafs. And now meafuring the Diameter of the
fifth dark Circle as accurately as I could, I found it the
fifth part of an Inch precifely. This meafure was taken
with the points of a pair of Compaffes on the upper fur-
face on the upper Glafs, and my Eye was about eight
or nine Inches diftance from the Glafs, almofi: perpen-
dicularly over it, and the Glafs was '^ of an Inch thick,
and thence it is eafy to coUeft that the true Diameter
of the Ring between the Glaflcs was greater than its
mcafured Diameter above the Glaffes in the proportion
of 80 to 79 or thereabouts, and by confequence equal
to ^ parts of an Inch, and its true Semi-diameter equal
to ^ parts. Now as the Diameter of the Sphere ( 1 82 In-
ches) is to the Semi-diameter of this fifth dark Ring
( ~ parts of an Inch ) fo is this Semi-diameter to the
thicknefs of the Air at this fifth dark Ring ; which is
therefore ^7^, or ,^^ parts of an Inch, and the fifth
part thereof; viz. the -^ly^^^ P^'irt of an Inch, is the
thicknefs of the Air at the firfi of thefe dark Rings.

The fame Experiment I repeated with another dou-
ble convex Objed-glafs ground on both fides to one and
the fame Sphere. Its Focus was diftant from it 168^
Inches, and therefore the Diameter of that Sphere was
184. Inches. This Glafs being laid upon the fame
plain Glafs, the Diameter of the fifth of the dark
Rings, when the black Spot in their center appeared
plainly without prefling the Glaffes, was by the mea-
fure of the Compaffes upon the upper Glafs ^ parts
of an Inch, and by confequence between the Glaffes it
was g-^y. For the upper Glafs was I of an Inch thick,

Bb and

[lo]

and iny Eye was diftant from it 8 Inches. And a third
proportional to half this from the Diameter of the
Sphere is ^^^ parts of an Inch. This is therefore the
thicknefs of the Air at this Ring, and a fifth part there-
of, viz. the issT-^th part of an Inch is the thicknefs there-
of at the firft of the Rings as above.

I tryed the fame thing by laying thefe Object-GlafTcs
upon flat pieces of a broken Looking-glafs, and found
the fame mcafures of the Rings : Which makes me
rely upon them till they can be determined more ac-
curately by Glaffes ground to larger Spheres, though
in fuch GlalTes greater care muft be taken of a true
plain.

Thefe Dimenlions were taken when my Eye was
placed almofl: perpendicularly over t]ie Glailes, being ,
about an Inch, or an Inch and a quarter, dillant from
the incident rays, and eight Inches diftant from the
Glafs ; fo that the rays were inclined to the Glafs in an
Angle of about 4. degrees. Whence by the following
Obfervation you will underftand, that had the rays
been perpendicular to the Glaffes, the thicknefs of the
Air at thefe Rings would have been lefs in the propor-
tion of the Radius to the fecant of 4. degrees, that is of
1 0000. Let the thickneffes found be therefore dimi-
nilhed in this proportion, and they will become 5^ and
i^, or ( to ufe the neareft round number ) the g^th
part of an Inch. This is the thicknefs of the Air at the
darkeft part of the firft dark Ring made by perpendi-
cular rays, and half this thicknefs multiplied by the
progreffion,i,^,5,7,9, i i,i5)'<r. gives the thickneffes of the
Air at the moft luminous parts of all the brighteft
Rings,, mz. j^, .-^„ j^, 7^,, }^c. their arithmetical

means

[II]

means ,-^, 77^, ttsst, ^(^' being its thickneffes at the
darkeft parts of all the dark ones.

O B S. VII.

The Rings were leaft when my Eye was placed per-
pendicularly over the Glaffes in the Axis of the Rings :
And when I viewed them obliquely they became big-
ger, continually fwelling as I removed my Eye further
from the Axis. And partly by meafuring the Diameter
of the fame Circle at feveral obliquities of my Eye,
partly by other means, as alfo by making ufe of the
two Frifms for very great obliquities. I found its Dia-
meter, and confequently the thicknefs of the Air at its
perimeter in all thofe obliquities to be very nearly in the
proportions exprefled in this Table.

Angle

of In-

Angle of Re-

Diameter of

Thicknefs of

ctdence

on the

fraBion into

the King.

the Air.

Air.

the Air.

deg.

min.

00 00

10 00

10:7

I Oil

4-5

20 00

JOj

IO-:

30 00.

icf

ii-i

40 00

50 00

I2I

^5r

60 00

65 00

15;-

^3?

70 00

I6-:

28,^

75 00

19^

^7 ■

80 00

^^1

5^4-

85 00

84if

1 1

90 00

122-[

Bb 2

[12]

In the two firft Columns are exprefled the obliquities
of the incident and emergent rays to the plate of the
Air, that is, their angles of incidence and refradion. In
the third Column the Diameter of any coloured Ring
at thole obliquities is exprefled in parts, of which ten
conftitute that Diameter when the rays are perpendicu-
lar. And in the fourth Column the thicknefs of the Air
at the circumference of that Ring is exprefled in parts
of which alfo ten conilitute that thicknefs when the rays
are perpendicular.

And from thefe meafures I feem to gather this Rule :
That the thicknefs of the Air is proportional to the fe-
cant of an angle, whole Sine is a certain mean propor-
tional between the Sines of incidence and retraction.
And that mean proportional, fo far as by thefe meafures
I can determine it, is the hrft of an hundred and fix
arithmetical mean proportionals between thofe Sine?
counted from the Sine of refradion when the refra-
ction is made out of the Glafs into the plate of Air, or
from the Sine of incidence when the refraction is
made out of the plate of Air into the Glafs.

O B S. VIII.

The dark Spot in the middle of the Rings increafed
alfo by the obliquation of the Eye, although almoft in-
fenfibly. But if infteadoftheObjedt-Glafles thePrifms
were made ufe of, its increafe was more manifefl: when
viewed fo obliquely that no Colours appeared about it.
It was leafl: when the rays were incident moft obliquely
on the interjacent Air, and as the obliquity decreafed
it increafed more and more until the coloured Rings ap-
peared.

[13],

peared, and then decreafed again, but not fo much as
it increafed before. And hence it is evident, that the
tranfparency was not only at the ablblute contact of the
Glafles, but alfo where they had fome Uttle interval.
I have Ibmetimes obferved the Diameter of that Spot to
be between half and two fifth parts of the Diameter of
the exterior circumference of the red in the firft cir-
cuit or revolution of Colours when viewed almoft per-
pendicularly ; whereas when viewed obliquely it hath
wholly vanillicd and become opake and white like the
other parts of the Glafs ; whence it may be colleded
that the Giafles did then fcarcely, or not at all, touch
one another, and that their interval at the perimeter
of that Spot when viewed perpendicularly was about a
fifth or fixth part of their interval at the circumference
of the laid red.

O B S. IX.

By looking through the two contiguous Objed:-
Glanes, 1 found that the interjacent Air exhibited Rings
of Colours, as well by tranfmitting Light as by reflect-
ing it. The central Spot was now white, and from it
the order of the Colours were yellowifh red ; black ;
violet, blue, white, yellow, red; violet, blue, green,
yellow, red, '^c. But thefe Colours were very faint
and dilute unlefs when the Light was trajeded very
obliquely through the Glaffes : For by that means they
became pretty vivid. Only the firft yellowilTi red, like
the blue in the fourth Obfervation, was fo little and
feint as fcarcely to be difcerned. Comparing the co-
loured Rings made by reflexion , with thefe made by

tranf-

[H'J

tranfmiflion of the Light ; I found that white was op-
polite to black, red to blue, yellow to violet, and green
to a compound of red and violet. That is, thofe parts
of the Glafs were black when looked through, which
when looked upon appeared u^hite, and on the con-
trary. And To thofe which in one cafe exhibited blue,
did in the other cafe exhibit red. And the like of the
Fig. 5. other Colours. The manner you have reprefented in
the third Figure, where AB, CD, are the lurfaces of
the Glaffes contiguous at E, and the black lines be-
tween them are their diiiances in arithmetical progref-
iion, and the Colours written above are feen by re-
flected Light, and thofe below by Light tranfmitted.

O B S. X.

Wetting the Objed-Glaffes a little at their edges,
the w^ater crept in flowly between them, and the Cir-
cles thereby became lefs and the Colours more faint :
Infomuch that as the water crept along one half of
them at which it firft arrived would appear broken off
from the other half, and contracted into a lefs room.
By meafuring them I found the proportions of their
Diameters to the Diameters of the like Circles made by
Air to be about feven to eight, and confequently the in-
tervals of the Glaffes at like Circles, caufed by thole
two mediums Water and Air,are as about three to four.
Perhaps it may be a general Rule, That if any other
medium more or lefs denfe than water be compreffed
between the Glaffes, their intervals at the Rings caufed
thereby will be to their intervals caufed by interjacent

Air,

J '5 ]

Air, as the Sines are which meafure the refradion made
out of that medium into Air.

O B S. XI.

When the water was between the Glafles, if I pref-
fed the upper Glafs varioufly at its edges to make the
Rings move nimbly from one place to another, a little
wliite Spot would immediately follow the center of
them, which upon creeping in of the ambient water
into that place would prefently vanifli. Its appearance
was fuch as interjacent Air would have caufed, and it
exhibited the lame Colours. But it was not Air, for
where any bubbles of Air were in the water they would
not vanifh. The reflexion mull have rather been caufed
by a fubtiler medium, which could recede through the.
Glaffes at tlie creeping in of the water.

O B S. XII.

Thefe Obfervations were made in the open Air. But
further to examin the effeds of coloured Light fal|>ing
on the Glafles, I darkened the Room, and viewed them
by reflexion of the Colours of a Prifm call on a Sheet
of white Paper, my Eye being fo placed that I could
fee the coloured Paper by reflexion in the Glafles, as
in a Looking-glafs. And by this means the Rings be-
came diftind:er and viflble to a far greater number than
in the open Air. I have fometimes feen more than
twenty of them, whereas in the open Air I could not
difcern above eight or nine.

oBs:.

[Id]

O B S. XIII.

Appointing an affillant to move the Prifm to and
fro about its Axis, that all the Colours might fuccef-
fiveiy fall on that part of the Paper which I faw by
reflexion from that part of the GlafTes, where the Cir-
cles appeared, fo that all the Colours might be fuccef-
fively refle61:ed from the Circles to my Eye whilft I held
it immovable, I found the Circles which the red Light
made to be manifeftly bigger than thole which were
made by the blue and violet. And it was very plea-
lant to fee them gradually fwell or contrad: according
as the Colour of the Light was changed. The inter-
val of the Glaffes at any of the Rings when they wTre
made by the utmoft red Light, was to their interval at
the lame Ring when made bythe utmoft violet, greater
than as ^ to 2, and lefs than as 1 5 to 8,by the moft of my
Obfervations it was as 14. to 9. And this proportion
feemed very nearly the fame in all obliquities of my
Eye ; unlefs when two Prifms were made ufe of inftead
of the Objed-Glaffes. For then at a certain great
obliquity of my Eye, the Rings made by the feveral
Colours feemed equal, and at a greater obliquity thole
made by the violet would be greater than the fame
Rings made by the red. The refraction of the Prifm
in this cafe caufing the moft refrangible rays to fall
more obliquely on that plate of the Air than the leaft
refrangible ones. Thus the Experiment fucceeded in
the coloured Light, which was fufBciently ftrong and
copious to make the Rings fenfible. And thence it
may be gathered, that if the moft refrangible and leaft

refran-

Ci7]

refrangibk rays had been copious enough to make the
Rings lenfible without the mixture of other rays, the
proportion which here was 14. to 9 would have been a
little greater, fuppofe 14. J or 14 Uo 9.

O B S. XIV.

Whilft the Prifni was turn'd about its Axis with an
uniform motion, to make all the feveral Colours fall
fucceffively upon the Object -Glaffes, and thereby to
make the Rings contract and dilate : The contrad:ion
or dilation of each Ring thus made by the variation of
its Colour was fwifteft in the red, and floweft in the
violet, and in the intermediate Colours it had inter-
mediate degrees of celerity. Comparing the quantity
of contraction and dilation made by all the degrees of
each Colour, I found that it was greateft in the red ;
lefs in the yellow, iHU lefs in the blue, and leaft in the
J violet. And to make as juft an eftimation as 1 could of the
'proportions of their contractions or dilations, 1 obferved
that the whole contradion or dilation of the Diameter
of any Ring made by all the degrees of red, was to that
of the Diameter of the fame Ring made by all the de-
grees of violet, as about four to three, or five to four, and
that when the Light was of the middle Colour between
yellow and green, the Diameter of the Ring was very
nearly an arithmetical mean between the greateft Dia-
meter of the fame Ring made by the outmoft red, and
the leaft Diameter thereof made by the outmoft violet :
Contrary to what happens in the Colours of the oblong
Spedrum made by the refradion of a Prifm, where the
red is moft contracted J the violet moft expanded, and

D d in

Ci8]

in the'midft of all tlie Colours is the confine of green
and blue. And hence 1 fecm to colled that the thick-
nefles of the Air between tlie Glaffes there, where the
Ring is fucceffively made by the limits of the five prin-
cipal Colours (red, yellow, green, blue, violet) in order
( that is, by the extreme red, by the limit of red and
yellow in the middle of the orange, by the limit of
yellow and green, by the limit of green and blue, by
the limit of blue and violet in the middle of the in-
digo, and by the extreme violet ) are to one another
very nearly as the fix lengths of a Chord which found
the notes in a fixth Major, /<?/, la^ mi^ fa^ fol^ la. But
it agrees fomething better with the Obfervation to fay,
that the thicknefles of the Air between the Glafi'es there,
where the Rings are fucceffively made by the limits of
the feven Colours, red, orange, yellow, green, blue, in-
digo, violet in order, are to one another as the Cube-
1 oots of the Squares of the eight lengths of a Chord,
which found the notes in an eighth , /o/, la^ fa^ fol^ la^
m^ fiij fol ; that is, as the Cube-roots of the Squares
df the Numbers, i , % |, ^ J, f, ■;!, f.

O B S. XV.

Thefe Rings were not of various Colours like thofe
iliade in the open Air, but appeared all over of that
prifmatique Colour only Vv'ith which they were illu-
minated. And by projeding the prifmatique Colours
imrhediately upon the Glaffes, I found that the Light
which fell on the dark Spaces which were between
the coloured Rings , was tranfmitted through the
Giafles without any variation of Colour. For on a

white

[19]

white Paper placed behind, it would paint Rings of
the fame Colour with thofe which were reflected, and
of the bignefs of their immediate Spaces. And from
thence the origin of thefe Rings is manifeft; namely.
That the Air between the Glaffes, according to its va-
rious thicknefs, is difpofed in fome places to retieiSt^
and in others to tranlmit the Light of any one Co-
lour (as you may fee reprefented in the fourth Figure ) pVg-, ^.
and in the fame place to reflect that of one Colour
where it tranfmits that of.anotlier.

O B S. XVL

The Squares of the Diameters of thefe Rings made
by any prifmatique Colour were in arithmetical pro-
greffion as in the fifth Obfervation. And the Diameter
of the fixth Circle, when made by the citrine yellow,
and viewed almoft perpendicularly, was about ^~ parts
of an Inch, or a little lefs, agreeable to the fixth Ob-
fervation.

The precedent Obfervations were made with a rarer
thin medium, terminated by a denier, fuchas was Air
or Water comprefifed between two Glaffes. In thofe
that follow are let down the appearances of a denfer
medium thin'd within a rarer, fuch as are plates <^
Mufcovy-glafs, Bubbles of Water, and fome other thin
fubrtances terminated on all fides with Air.

Dd a OBS,

[20]

O B S. XVIL

. If a Bubble be blown with Water firft made tenacious
by diffolving a little Soap in it, 'tis a common Obler-
vation, that after a while it will appear tinged with a
great variety of Colours. To defend thefe Bubbles
from being agitated by the external Air (whereby their
Colours are irregularly moved one among another, fo
that no accurate Obfervation can be made of them,) as
foon as I had blown any of them 1 covered it with a
clear Glafs, and by that means its Colours emerged in
a very regular order, like fo many conccntrick Rings
incompaffing the top of the Bubble. And as the
Bubble grew thinner by the continual fubliding of the
Water, thefe Rings dilated tlowly and over-fpread the
whole Bubble, delcending inorder to the bottom of it,
where they vanillied fuccellively. In the mean while,
after all thte Colours were emerged at the top, there
grew in the Center of the Rings a fmall round black.
Spot, like that in the firft Obfervation, which conti-
nually dilated it felf till it became fometimes more than
'- or I of an Inch in breadth before the Bubble broke.
At firft I thought there had been no Light refleded from
the Water in that place, but obferving it more cu-
rioufly, 1 faw within it feveral fmaller round Spots,
which appeared much blacker and darker than the reft,
whereby 1 knew that there was fome reflexion at the
other places which were not fo dark as thole Spots.
And by further tryal 1 found that 1 could fee the Images
of fome things (as of a Candle or the Sun ) very fliint-
ly refledled, not only from the great black Spot, but

alio

[21]

alfo from the little darker Spots which were with-
in it.

Befides the aforefaid coloured Rings there would
often appear fmall Spots of Colours, afcending and de-
fcending up and down the (ides of the Bubble, by reafon
of fome inequalities in the fubliding of the Water.
And fometimes fmall black Spots generated at the fides
would afcend up to the larger black Spot at the top of
the Bubble,^ and unite with it,

O B S. XVIIL

Becaufe the Colours of thefe Bubbles were more ex-
tended and lively than thofe of the Air thin'd between
two Glafll's, and fo more ealy to to. dilHnguifhed , I
fhal'l here give you a furtlier defcription; of their order^
as they were obferved in viewing them, by reflexion of
the Skies when of a white Colour, whilft a black Sub-
ftance was placed behind the Bubble. And they were
thefe, red, blue; red, blue; red, blue; red, green;,
red, yellow, green, blue, purple ; red, yellow, green,,
blue, violet ; red, yellow, white, blue, black.

The three firil Succeffions of red and blue were very
dilute and dirty, efpecially the firft^ where the red.
feemed in a manner to be white^ Among thefe there
was fcarce any other Colour fenfible befides red and
blue, only the blues ( and principally the fccond blue)
inclined a little to green.

The fourth red was alfo dilute and dirty, but not
fo much as the former three ; after that fucceeded little
or no yellow, but a copious green, which at firll incli-
ned a little to yellow, and then became a pretty brifque

and:

[22]

and good willow green, and afterwards changed to a
bluilli Colour; but there fucceeded neither blue nor
violet.

The fifth red at firft inclined very much to purple,
and afterwards became more bright and brifque, but
yet not very pure. This was fucceeded with a very
bright and intenfe yellow , which was but little in
quantity, and foon changed to green : But that green
was copious and fomething more pure, deep and lively,
than the former green. After that followed an excel:-
lent blue of a bright sky-colour, and then a purple,
which was lefs in quantity than the blue, and much
inclined to red.

The fixth Red was at firft of a very fair and lively
Scarlet, and foon after of a brighter Colour , being
very pure and brifque , and the beft of all the
reds. Then after a lively orange followed an intenfe
bright and copious yellow, which was alio the beft
of all the yellows, and this changed firft to a greeniOli
yellow, and then to a greenilh blue ; but the green;
between the yellow and the blue, was very little and
dilute, teeming rather a- greenilh white than a green.
The blu€ wiiich fucceeded became very good, and of a
very fair bright sky-colour, but yet fomething inferior
to the former blue _; and the violet was intenfe and
deep with little or no rednefs in it. And lefs in quan-
tity than the blue.

In the laft red appeared a tindure of fcarlet next
to violet, which foon changed to a brighter Colour,
inclining to an orange ; and the yellow which followed
was at firft pretty good and lively , but afterwards it
grew more dilute, until by degrees it ended in perfe<a

wliite-

whitenefs. And this whitenefs, if the Water was very
tenacious and well-tempered, would flowly Ipread and
dilate it felf over the greater part of the Bubble 3 con-
tinually growing paler at the top, where at length it
would crack in many ]:)laces, and thofe cracks, as they
dilated, would appear of a pretty good, but yet obfcure
and dark sky-colour; the white between the blue Spots
diminifhing, until it refembled the threds of an irre-
gular Net-work, and foon after vaniflied and left all
the upper part of the Bubble of the laid dark blue
Colour. And this Colour, after the aforefaid manner,
dilated it felf downwards , until fometimes it hath
overfpread the whole Bubble. In the mean while at
the top, which was of a darker blue than the bottom,
and appeared alfo full of many round blue Spots, fome-
thing darker than the reft , there would emerge one
or more very black Spots, and within thofe other Spots
of an intenfer blacknefs, which I mentioned in the
former Obfervation ; and thefe continually dilated
themfelves until the Bubble broke.

If the Water was not very tenacious the black Spots
would break forth in the white, without any fenlible
intervention of the blue. And fometimes they would
break forth within the precedent yellow , or red, or
perhaps within the blue of tlje fecond order, before
the intermediate Colours had time to difplay them^
felves.

By this defcription you may perceive how great an
affinity thefe Colours have with, thofe of Air defcri-
bed in the fourth Obfervation, although fet down in
a contrary order, by reafon that they begin to appear
when the Bubble is thickeft , and are molt conve-
niently

niently reckoned trom the lowclt aiid thickeft part of
the Bubble upwards.

O B S. XIX.

Viewing in feveral oblique pofitions of my Eye
the Rings of Colours emerging on the top of the Bubble,
1 found that they were feufibly dilated by increaiing
the obliquity, but yet not fo much by far us thofe
made by thin'd Air in the feventh Obfervation. For
there they were dilated fo much as, when viewed
moft obliquely, to arrive at a part of the plate more
than twelve times thicker than that where th^^y ap-
peared when viewed perpendicularly; whereas in this
cafe the thicknefs of the Water, at which they arrived
when viewed moft obliquely, was to that thicknefs
which jexhibited them by perpendicular rays, fome-
thing lets than as 8 to 5. By the beft of myObfervations
it was between 15 and 15^ to 10, an increafe about
a 4 times lefs than in the other cafe.

Sometimes the Bubble would become of an uniform
thicknefs all over, except at the top of it near the black
Spot, as I knew, becaufe it would exhibit the fame
appearance of Colours in all potitions of the Eye. And
then the Colours which were feen at its apparent cir-
cumference by the obliquell rays, would be ditferent
from thofe that were feen in other places, by rays lefs
oblique to it. And divers Spectators might fee the
i;ime part of it of differing Colours, by viewing it at
very differing obliquities. Now obferving how much
the Colours at the fame places of the Bubble, or at di-
vers places of equal thicknefs , were varied by the

feveral

[25 3

feveral obliquities of the rays ; by the alTiftance of the
4th, 14th, 1 6th and i8th Obfervations, as they are
hereafter explained, I collect the thicknefs of the Water
requifite to exhibit any one and the fame Colour, at fe-
veral obliquities , to be very nearly in the proportion
€xprefled in this Table.

Incidence on
the Water.

Refraction in-
to the Water.

Thickttefs of
the Witter.

deg. min.
00 00

15 00

II II

■0;

50 00

12 I

IO-;

45 00
60 00

75 '^o
90 00

g2 1
40 30
46 25

48 55

'5f 1

In the two firft Columns are expreffed the obliqui-
ties of the rays to the fuperficies of the Water, that
is, their Angles of incidence and refraction. Where
I fuppofe that the Sines which meafure them are in
round numbers as 5 to 4, though probably the diffo=
lution of Soap in the Water , may a little alter its
refractive Vertue. In the third Column the thicknefs
of the Bubble, at which any one Colour is exhibited
in thofe feveral obliquities, is expreft in parts,of which
ten conftitute that thicknefs when the rays are perpen=
dicular.

I have fometimes obferved, that the Colours which
arife on polillied Steel by heating it, or on Bell-metal^
and fome other metalline fubftances, when melted and

E e poured

[26]

'poured on the ground , where they may cool in the
-open Air, have, like the Colours of Water-bubbles,
"been a iittle changed by viewing them at divers ob-
iiquities, and particularly that a deep blue, or violet^
■when vievved very obliquely, hath been changed to a
deep red. But the changejs of thefe Colours are not lb
great and feniible as of thofe made by Water. For the
Scoria or vitrified part of the Metal, which moll Me-
tals when heated or melted do continually protrude,
and fend out to their furface, and which by covering
the Metals in form of a thin glafly skin, caufes thefe
Colours, is much denfer than Water ; and I find that
the change made by the obliquation of the Eye is leail
in Colours of the denfeft thin fubftances.

O B S. XX.

As in the ninth Obfervation, fo here^ the Bubble, by
tranfmitted Light, appeared of a contrary Colour to
that which it exhibited by reflexion. Thus when the
Bubble being looked on by the Light of the Clouds j:e-
fle6led from it, fecmed red at its apparent circumfe-
rence, if the Clouds at the fame time, or immediately
after, were viewed through it, the Colour at its cir-
cumference would be blue. And, on the contrary,
when by refleded Light it appeared blue, it would ap-
pear red by tranfmitted Light.

O B S. XXL

By wetting very thin plates of Mufcovy-glafs, whofe
thinnefs made the like Colours appear, the Colours

became

[27]

became more faint and languid ; efpecially by wetting
the pktes on that fide oppofite to the Eye : But I eould
not perceive any variation of their fpecies. So then
the thicknefs of a plate requifite to produce any Co-
lour, depends only on the denfity of the plate, and
not on that of the ambient medium: And hence, by the
loth and i6th Obfervations, may be known the thick-
nefs which Bubbles of Water, or Plates of Mufcovy-
glafs, or other fubftances, have at any Colour pro-
duced by them.

O B S. XXII.

A thin tranfparent Body, which is denfer than its
ambient medium, exhibits more brifque and vivid Co-
lours than that which is fo much rarer ; as I have
particularly obferved in the Air and Glafs. For blow-
ing Glafs very thin at a Lamp-furnace, thofe plates
incompafled with Air did exhibit Colours much
more vivid than thofe of Air made thin between two
Glafles.

O B S. XXIIL

Comparing the quantity of Light refiefted from the
feveral Rings, I found that it was mod copious from
the firft or inmoft, and in the exterior Rings be-
came gradually lefs and lefs. Alfo the whitenefs of
the firft Ring was ftronger than that refleded from
thofe parts of the thinner medium which were with-
out the Rings ; as I could manifeftly perceive by view-
ing at a diftance the Rings made by the two Obje6l-

Ee 3 Glaffes,

[28]

Glaffes; or by comparing two Bubbles of Water blown
at diftant times, in the firft of which the whitenefs
appeared, which fucceeded all the Colours, and in
the other, the whitenefs w^hich preceded them all.

O B S. XXIV.

When the two Objed-Glaifes were lay'd upon one
another, fo as to make the Rings of the Colours ap-
pear, though with my naked Eye 1 could not difcern
above 8 or 9 of thofe Rings, yet by viewing them,
through a Prifm I have feen a far greater multitude,
infomuch that 1 could number more than forty, betides
many others, that were fo very fmail and clofe toge^
ther, that 1 could not keep my Eye fteddy on them
feverally fo as to number them, but by their extent I have
Ibmetimes eftimated them to be more than a hundred.
And 1 believe the Experiment may be improved to the
difcovery of far greater numbers. For they feem to
be really unlimited, though vilible only fo far as they
can be feparated by the refraction, as 1 fhall hereafter
explain.

But it was but one fide of thefe Rings, namely, that
towards which the refradion was made, which by that
refraction was rendered diftinCt, and the other fide be-
came more confufed than when viewed by the naked
Eye, infomuch that there 1 could not difcern above
one or two, and fometimes none of thofe Rings , of
which I could difcern eight or nine with my naked
Eye. And their Segments or Arcs, which on the
other fide appeared lb numerous,, for the moft part

exceeded

[29]

exceeded not the third part of a Circle. If the Re-
fraction was very great, or the Prifm very diftant from
the Objed-Glafles, the middle part ofthofe Arcs be-
came alfo confufed, fo as to difappear and conftitute an
even vvhitenefs, whilft on either fide their ends, as alfo
the whole Arcs furtheft from the center, became di-
ftinder than before, appearing in the form as you fee
them defigned in the fifth Figure. Ftg. 5.

The Arcs, where they feemed difiiindefi:, were only
white and black fucceflively, without any other Co-
lours intermixed. But in other places there appeared
Colours, whofe order was inverted by the refraction
in fuch manner, that if I firfl: held the Prifm very near
the ObjeCt-GlafTes , and then gradually removed it
further otf towards my Eye, the Colours of the ad,
3d, 4.th, and following Rings llirunk towards the white
that emerged between them , until they wholly va-
nilhed into it at the middle of the Arcs, and after-
wards emerged again in a contrary order. But at
the ends of the Arcs they retained their order un-
changed.

I have fometimes fo lay'd one Obje£t-Glafs upon
the other, that to the naked Eye they have all over
feemed uniformly white, without the leaft appearance •
of any of the coloured Rings ; and yet by viewing
them through a Prifm, great multitudes ofthofe Rings
have difcovered themfelves. And in like manner plates
of Mufcovy-glafs , and Bubbles of Glafs blown at a
Lamp-furnace, which were not fo thin as to exhibit
any Colours to the naked Eye, have through the Prifm
exhibited a great variety of them ranged irregu-
larly up and down in the form of waves. And fo

Bubbles

C 30 ]

Bubbles of Water, before they began to e^^hibit their
Colours to the naked Eye of a By-ftander, have ap-
peared through a Prifm, girded about with many pa-
rallel and horizontal Rings- to produce which effed,
it was neceflary to hold the Prifm parallel, or very
nearly parallel to the Horizon, and to difpofe it fo
that the rays might be refracted upwards.

THE

SECOND BOOK

O ¥

O P T I C K S.

of the

PART II.

Remarks u^on the foregoing OSfervations.

'Aving given my Obfervations of thele Colours,,

before I make ufe of them to unfold the Caufes

o? the Colours of natural Bodies, it is convenient that
by the fimpleft of tliem, fuch as are the ad, ^d, /j-th,
9th, lath, 18th, aoth, and a4th , I firft explain the
more expounded. And firft to Ihew how the Colours
in the fourth and eighteenth Obfervations are produ-
c-ed, let there be taken in any right line from the point
y, the lengths YA, YB, Y C, YD, YE, YF, YG.Fig-^.
Y H, in proportion to one another, as the Cube- roots
pf the Squares of the numbers, {, ^, i,^, J, |, |, i, where-
by the lengths of a mufical Chord to found all the Notes
in an Eighth are reprefcntad; that is, in the propor-'
fion of the numbers 6^00, 6814, 7114, 7631, 8255,
^§559 9H.h ^^^^^' -^id at the points A, B, C, D,,

I 32]

E, F, G, H, let perpendiculars Aa^ )i$^l^c. beere(fl:ed,
by whole intervals the extent of the feveral Colours
let underneath againft them, is to be reprefented. Then
divide tlie line A x m fuch proportion as the numbers
I, a, ^, 5, 6, 7, 9, lo, 1 1, If^c. fet at the points of divi-
fion denote. And through thofe divifions from Y
draw lines i I, ^ K, 3 L, 5 M, 6 N, 7 0,i>r.

Now if A 2 be iuppofed to reprefent the thicknefs
of any thin traniparent Body , at which the outmoft
violet is moft copioufly reflected in the tirll Ring, or
Series of Colours, then by the i^th Obfervation H K,
will reprefent its thicknefs, at which the utmoft red
is moft copioufly reflected in the fame Series. Alfo
by the 5th and i6th Obfervations, A 6 and HN will
denote the thicknefles at which thofe extreme Colours
are moft copioufly reflated in the fecond Series, and
A I o and H Q the thicknefles , at which they are
moft copioufly refleded in the tliird Series, and fo on.
And the thicknefs at which any of the intermediate
Colours are reflected moft copioufly, will, according to
the 1 4.th Obfervation, be defined by the diftanceof the
line A H -from the intermediate parts of the lines a K,
6N, 10 Q, }sfc. againft which the names of thofe Co-
lours are written below.

But further, to define the latitude ofthefe Colours in
each Ring or Series, let A i defign the lealt thicknefs,
and A 3 the greateft thicknefs, at which the extreme
violet in the nrft Series is refleded, and let H I, and
H L, defign the like limits for the extreme red, and
let the intermediate Colours be limited by the inter-
mediate parts of the lines i I, and ^L, againft vvhih
the names of thofe Colours are written, and fo on : But

yet

yet with this caution, that the refle£l:ions be fuppofed
ih'ongeil at the intermediate Spaces, a K, 6 N, i o (i,^<r-,
and from thence to decreale gradually towards thefe li-
mits, I I, ^ L, 5 M, 7 O, }£^c. on either fide ; where
you muft not conceive them to be precifely limited,
but to decay indefinitely. And whereas 1 have affigned
the fame latitude to every Series, 1 did it, becaufe al-
though the Colours in the firft Series feem to be a little
broader than the refi:, by reafon of a fl:ronger reflexion
there, yet that inequality is fo infenfiblc as fcarcely to
be determined by Obfervation,

Now according to this defcription, conceiving that
the rays originally of feveral Colours are by turns re-
fleded at the Spaces 1 1 L ^, 5 M O 7, 9 P R 1 1 , isJ'r*
andtranfmitted at the Spaces AHIi,^LM5,70P9,
Isfc. it is eafy to know what Colour muft: in the open Air
be exhibited at any thicknefs of a tranfparent thin body.
For if a Ruler be applied parallel to A H, at that di-
ftance from it by which the thicknefs of the body is
reprefented, the alternate Spaces i IL ^, 5 MO jyWc,
which it crofleth will denote the refleded original Co-
lours, of which the Colour exhibited in the open Air
is compounded. Thus if the conft:itution of the green
in the third Series of Colours be defired, apply the
Ruler as you fee at f e'^f, and by its paffing through
fome of the blue at ^ and yellow at<^, as well as through
the green at ^, you may conclude that the green exhi-;
bited at that thicknefs of the body is principally con-
ftituted of original green, but not without a mixture
of fome blue and yellow.

Ff By

[34]

By this means you may know how the Colours from
the center of the Rings outward ought to fucceed in
order as they were deicribed in the 4th and iSthOb-
fervations. For if you move the Ruler gradually from
AH through all diftances, having paft over the firft
fpace which denotes little or no reHexion to be made
by thinneft fubftances, it will firft arrive at i the violet,
and then very quickly at the blue and green, which
together with that violet compound blue, and then at
the yellow and red , by whofe further addition that
blue is converted into whitenefs, which whitenefs con-
tinues during the tranfit of the edge of the Ruler from
I to 5, and after that by the fucceffive deficience of
its component Colours, turns firft to compound yellow^
and then to red, and laft of all the red ceafcth at L.
Then begin the Colours of the fecond Series, which
fucceed in order during the tranfit of the edge of the
Ruler from 5 to O, and are more lively than before,
becaufe more expanded and fevered. And for the
fame reafon, inftead of the former white there inter-
cedes between the blue and yellow a mixture of orange,,
yellow, green, blue and indico, all which together ought
to exhibit a dilute and imperfed green. So the Co-
lours of the third Series all fucceed in order ; firft, the
violet, which a little interferes with the red of the fe-
cond order, and is thereby inclined to a reddifh purple ;
then the blue and green , which are lefs mixed with
other Colours, and confequently more lively than be-
fore, efpecially the green: Then follows the yellow ,
fomeot which towards the green is diftind: and good, but
that part of it towards the fucceeding red, as alfo that
r^d is mixed with the violet and blue of the fourth Se-

riesj

[ ?5 3

lies, whereby' Various degrees of red very much incli-
ning to purple are compounded. This violet and blue,
which (hould fucceed this red, being mixed with, and
hidden in it, there fucceeds a green. And this at firft
is much inclined to blue, but foon becomes a good
green , the only unmixed and lively Colour in this
fourth Series. For as it verges towards the yellow, it
begins to interfere with the Colours of the fifth Series,
by whole mixture the fucceeding yellow and red are
very much diluted and made dirty, efpecially the yel-
low, which being the weaker Colour is fcarce able to
Ihew it felf. After this the feveral Series interfere more
and more, and their Colours become more and more
intermixed, till after three or four more revolutions
( in which the red and blue predominate by turns )
all forts of Colours are in all places pretty equally ben-
ded, and compound an even whitenefs.

And fince by the 1 5 th Obfervation the rays indued
with one Colour are tranfmitted, where thofe of ano'
ther Colour are relieded, the reafon of the Colours
made by the tranfmitted Light in the 9th and 20th Ob-
fervations is from hence evident.

If not only the order and fpecies of thefe Colours^
but alio the precife thicknefs of the plate, or thin body
at which they are exhibited, be defired in parts of an
Inch, that may be alfo obtained by affiftance of the 6th
or 1 6th Obfervations. For according to thofe Obferva-
tions the thicknefs of the thinned Air, which between
two Glafles exhibited the moft luminous parts of the
nrit iix ivings weie i^jsoo) itSoooj TtHoco) i7»coc) itSooo) 17S000 parrs or
an Inch. Suppofe the Light reflected moft copioufly
at thefe thicknefles be the bright citrine yellow, or con-

Ff 2 fine

fine of yellow and orange, and thefe thickneffes will
be^M, Gv, G^, Go, G^. And this being known, it is
eafy to "determine what thicknefs of Air is reprefented
by G^, or by any other diftance of the ruler from
AH.

But further, fince by the i oth Obfervation the thick-
nefs of Air was to the thicknefs of Water, which be-
tween the fame Glaffes exhibited the fame Colour, as
4 to ^, and by the aith Obfervation the Colours of
thin bodies are not varied by varying the ambient me-
dium ; the thicknefs of a Bubble of Water, exhibiting
any Colour, will be \ of the thicknefs of Air producing
th€ fame Colour. And fo according to the lame i oth
and aith Obfervations the thicknefs of a plate of
Glafs, whole refraction of the mean refrangible ray, is-
meafured by the proportion of the Sines ^i to 20,
may be f^ of the thicknefs of Air producing the fame
Colours ; and the hke of other mediums. I do not
affirm, that this proportion of ao to 51, holds in all
the rays ; for the Sines of other forts of rays have other
proportions. But the differences of thofe proportions
are fo little that I do not here confider them. On
thefe Grounds 1 have compofed the following Table,
wherein the thicknefs of Air, Water, and Glafs, at
which each Colour is moft intenfe and fpecifick, isex-
preiTed in parts of an Inch divided into Ten hundred
rhoufand equal parts.

Tht

C37]

The thichmfs of coloured Tlates and Tdrticles of

Their Colours of the'
firft Order,

Of the fecond Order,

''Very Black
Black
Beginning of

Black
Blue
White
Yellow
Orange
-Red

^Violet
Indico
Blue
Green
'< Yellow
Orange
Bright Red
.Scarlet

■^Purple
Indico
Blue
Of th& third Order, ^ Green

Yellow
Red
-^Bluifh Red

Bluifh Green
)Green
lYellowifli Green

Red

5Greenifli Blue
^Red

Of the fixth Order, j Greenifli Blue
. ^Red

Of thefeventhoSerjCreenirh Blue

^Ruddy White

Ofthe fourth Order,

Ofthe fifth Order,

Air.

W^4/^er.

GUfs.

■

1 0
3 I

i
4

1 0
3T

ii§

5."

Si
6

4i
5^

Il6

I2|

8i
9s^

lot

I Si

i^f

I2J

107

171

II?

iSi

191

12f

'1^

I Star

22-1

i6f

i4t

2?t
251

I7ii

157'
16%

27f

20f

i7t

-9

-u

i8f

20f

25t

?57

26t

22i

231

401

1<=>X

54i

291

52i
65

39i

44
48i

71
77

57i.

45t
491

Now.

Now if this Table be compared with the 6th Scheme,
you will there fee the conftitution of each Colour, as
to it& Ingredients, or the original Colours of which it
is compounded, ■ and thence be enabled to judge of its
intenfenefs or imperfection ; which may fuffice in ex-
plication of the 4.th and 1 8th Obfervations, unlefs it
be further dclired to delineate the manner how the Co-
lours appear, when the two ObjeLl-GlafTes are lay'd
upon one another. To do which, let there be dc-
Icribed a large Arc of a Circle, and a ftreight Line
w.hich may touch that Arc, and parallel to that Tan-
gent feveral occult Lines, at fuch diftances from it, as
the numbers fet againft the feveral Colours in the Table
denote. For the Arc, and its Tangent, will repreient
the fuperficies of the Glaffes terminating the interjacent
Air; and the places where the occult Lines cut the
Arc will fhow at what diftances from the Center, or
Point of contad, each Colour is refleded.

There are alio other ufes of this Table : For by its
affiftance the thicknefs of the Bubble in the 1 9th Ob-
fervation was determined by the Colours which it ex-
hibited. And fo the bignefs of the parts of natural
Bodies may be conjectured by their Colours, as fhall be
hereafter fhewn. Alfo, if two or more very thin plates
be lay'd one upon another, fo as to compofe one plate
equalling them all in thicknefs, the reiulting Colour
may be hereby determined. For inftance, Mr. Hook in
his Mijcrografhia obferves, that a faint yellow plate of
Mufcovy-glafs lay'd upon a blue one, conftituted a very
deep purple. The yellow of the firft Order is a faint
one, and the thicknefs of the plate exhibiting it, ac-
cording to the Table is 4|, to which add 9, the thick-
nefs

[39]

nefs exhibiting blue of the fecond Order, and the fum
will be i^f, which is the thicknefs exhibiting the
purple of the third Order.

To explain, in the next place, the Circumftances of
the ^d and 3d Obfervations ; that is, how the Rings of
the Colours may ( by turning the Prifms about their
common Axis the contrary way to that exprefled in
thofe Obfervations) be converted into white and black
Rings, and afterwards into Rings of Colours again, the
Colours of each Ring lying now in an inverted order; it
muft beremembred, that thofe Rings of Colours are di-
lated by the obliquation of the rays to the Air which
intercedes the GlalTes, and that according to the Table
in the 7th Obfervation, their dilatation or increafe of
their Diameter is moft manifeft and fpeedy when they
are obliqueft. Now the rays of yellow being more re-
frafted by the firft fuperhcies of the faid Air than thofe
of red, are thereby made more oblique to the fecond fu-
perhcies, at which they are reflected to produce the co-
loured Rings, and confequently the yellow Circle in each
Ring will be more dilated than the red; and the excels of
its dilatation will be fo much the greater, by how much
the greater is the obliquity of the rays, until at lall it be-
come of equal extent with the red of the fame Ring. And
for the fame reafon the green, blue and violet, will be alfo
fo much dilated by the ftill greater obliquity of their
rays, as to become all very nearly of equal extent with
the red, that is, equally diftant from the center of the
Rings. And then all the Colours of the fame Ring
muft be coincident, and by their mixture exhibit a
white Ring. And thefe white Rings muft have black
and dark Rings between them , becaufe they do not

fpread

.[40]

ipiread and interfere with one another as before. And
for that reafon aUb they muft become diftinfter and vi-
fible to f.ir greater Numbers. But yet the violet being
obliqueft will be fomething more dilated in proportion
to its extent then the other Colours, and fo very apt to
appear at the exterior verges of the white.

Afterwards, by a greater obliquity of the rays, the
violet and blue become more fenlibly dilated than the
red and yellow, and fo being further removed from the
center of the Rings, the Colours muft emerge out of the
white in an order contrary to that which they had be-
fore, the violet and blue at the exterior limbs of each
Ring,and the red and yellow at the interior. And the vio-
let, by reafon of the greateft obliquity of its rays, being
in proportion moft ot all expanded, will fooneft appear
at the exterior limb of each white Ring, and become
more confpicuous than the reft. And the leveral Series
of Colours belonging to the feveral Rings, will, by
their unfolding and Ipreading, begin again to interfere,
and thereby render the Rings lefs diftin^t, and not vifi-
ble to fo great numbers.

If inftead of the Prifms the Objed-glafles be made
ufe of, the Rings which they exhibit become not white
and dilHnCt by the obliquity of the Eye, by reafon that
the rays in their paflage through that Air which inter-
cedes the Glaffes are very nearly parallel to thofe Lines
in which they were firft incident on the Glaffes, and con-
fequently the rays indued with feveral Colours are not
inclined one more than another to that Air, as it hap-
pens in the Prifms.

There is yet another circumftance of thefe Experiments
to be conlidered, and that is why the black and white

Rings

C40

Rings which when viewed at a diftance appear diftind,
iTiould not only become confuted by viewing them near
at hand , but aUb yield a violet Colour at both the
edges of every white Ring. And the reafon is, that the
rays which enter the Eye at feveral parts of the Pupil,
have feveral obliquities to the Glaffes, and thofe which
are moft oblique, if confidered apart, would reprefent
the Rings bigger than thofe which are the leaft oblique.
Whence the breadth of the perimeter of every white
Ring is expanded outwards by the obliqueft rays,
and inwards by the leaft oblique. And this expanfion
is fo much the greater by how much the greater is the
difference of the obliquity • that is, by how much the
Pupil is wider, or the Eye nearer to the Glafles. And
the breadth of the violet muft be moft expanded, be-
caufe the rays apt to excite a fenfation of that Colour
are moft oblique to a fecond, or further fuperficies of
the thin'd Air at which they are refle6ted, and have
alfo the greateft variation of obliquity , which makes
that Colour fooneft emerge out of the edges of the
white. And as the breadth of every Ring is thus aug-
mented, the dark intervals muft be diminiflied, until
the neighbouring Rings become continuous, and are
blended, the exterior lirft, and then thofe nearer the
Center , fo that they can no longer be diftinguifti'd
apart, but feem to conftitute an even and uniform
whitenels.

Among all the Obfervations there is none accompa-
nied with fo odd circumftances as the 24.th. Of thofe
the principal are, that in thin plates , which to the
naked Eye feem of an even and uniform tranfparent

G g white-

[42]

whitenefs, without any terminations of fhadows, the
refraction of a Prifm fhould make Rings of Colours ap-
pear, whereas it ufually makes Obje<!;ts appear coloured
only there where they are terminated with fhadows, or
have parts unequally luminous; and that it fhould make
thofe Rings exceedingly diftinft and white, although
it ufually renders Objects confufed and coloured. The
caufe of thefe things you will underftand by confidering,
that, all the Rings of Colours are really in the plate,
when viewed with the naked Eye, although by reaibn
of the great breadth of their circumferences they To
much interfere and are blended together,that they ieeni
to conftitute an even whitenefs. But when the rays
pafs through the Prifm to the Eye, the orbits of the
leveral Colours in every Ring are refracted, fome more
than others, according to their degrees of refrangibility :
By which ixieans the Colours on one fide of the Ring
(that is on one fide of its Center) become more unfolded
and dilated, and thofe on the other fide more compli-
cated and contracted. And where by a due refraction
they are fo much contracted, that the fevral Rings be-
come narrower than to interfere with one another, they
mult appear diftinCt, and alfo white, if the conftituent
Colours be fo much contracted as to be wholly coincident.
But, on the other fide, where the orbit of every Ring
is made broader by the further unfolding of its Co-
lours, it muft interfere more with other Rings than
before, and^ fo become lefs diftinCt.

To explain this a little further, fuppofe the concen-

f^ff n^ trick Circles A V, and' BX, reprefent the red and violet

of any order, which, together with the intermediate

Colours,

[433

Colours, coiiftitute any one of thefe Rings. Now thefe
being viewed through a Prifnl, the violet Circle B X,
will by a greater rcfradion be further tranflated from
its place than the red A V, and fo approach nearer to
it on that fide, towards which the refradions are made.
For inftance, if the red be tranflated to av^ the violet
may be tranflated to ^ x, fo as to approach nearer to it
at X than before, and if the red be further tranflated
to a V, the violet may be [o much further tranflated to
b X as to convene with it at x, and if the red be yet
further tranflated to * i', the violet may be ftill fo much
further tranflated to 3 ? as to pafs beyond it at ?, and
convene with it at e and/. And this being underftood
not only of the red and violet, but of afl the other in-
termediate Colours, and alfo of every revolution of
thofe Colours, you wiU eafily perceive how thofe of the
fame revolution or order, by their nearnefs ^t xv and
'^^ ?, and their coincidence at xv, e and/, ought to con-
ftitute pretty diftindi Arcs of Circles, efpecially at x v,
or at e and /, and that they wifl appear feverally at
X -zr, and at x v exhibit whitenefs by their coincidence,
and again appear feveral at '^^ ?, but yet in a contrary
order to that which they had before, and ftiU retain
beyond e and f. But, on the other lidfe, at a-^, ab^
or * z^, thefe Colours mufl: becoine much more confu-
fed by being dilated and fpread fo, as to interfere with
thofe of other Orders. And the fame confufiori will
happen at i^ ? between e and/, if the refraction ht very
great, or the Prifm very diftant from the Obje<^-Gkfles :
In which cafe no parts of the Rings will be feen, fav6
only two little Arcs at e and/, whofediftarice from ont

Gg 2 another.

[44]

another will be augmented by removing the Prifin
ftill further from the Objedt-Glaffes : And thefe little
Arcs murt be diftindeft and whiteft at their middle, and
at their ends, where they begin to grow confufed they
muft be coloured. And the Colours at one end of
every Arc muft be in a contrary order to thofe at the
other end, by reafon that they crofs in the interme^
diate white ; namely their ends, which verge towards
'^ ?, will be red and yellow on that iide next the Cen-
ter,, and blue and violet on the other fide. But their
other ends which verge from '^ s will on the contrary
be blue and violet on that fide towards the Center, and
on the other fide red and yellow.

Now as all thefe things follow from the Properties
of Light by a mathematical way of reafoning, fo the
truth of them may be manifefted by Experiments. For
in a dark room , by viewing thefe Rings through a
Prifm, by reflexion of the feveral prifmatique Colours,,
which an affiftant caufes to move to and fro upon a
Wall or Paper from whence they are reflected, whilft
the Spectator's Eye, the Prifm, and the Objed-Glaflfes
(as in the 13th Obfervation) are placed fteddy : the
pofition of the Circles made fucceffively by the feveral
Colours, will be found fuch, in refpect of one another,
as 1 have defcribed in the Figures ahxv^ or abxv,
or »/3|T. And by the fame method the truth of
the Explications of other Obfervations may be exa-
mined.

By what hath been faid the like PhBenomina of
Water, and thin plates of Glafs may be underftood.
But in fmall fragments of thofe plates, there is this

further

[45]

further obfervable, that where they lye flat upon a
Table and are turned about their Centers whilft they are
viewed through a Prifm , they will in fome poftures-
exhibit waves of various Colours, and fome of them ex-
hibit thefe waves in one or two portions only, but the
moft of them do in all portions exhibit them, and make
them for the moft part appear almoft all over the plates.
The reafon is, that the fuperficies of fuch plates are not
even, but have many cavities and fwellings, which how
Ihallow foever do a little vary the thicknefs of the
plate. For at the feveral fides of thofe cavities, for
the reafons newly defcribed, there ought to be produ-
ced waves in feveral poftures of the Prifm. Now though-
it be but fome very fmall, and narrower parts of the
Glafs, by which thefe waves for the moft part are cau-
led, yet they may feem to extend themfelves over the
whole Glafs, becaufe from the narrowcft of thofe parts
there are Colours of feveral Orders that is of feveral
Rings, confufedly reflected, which by refradion of the
Prifm are unfolded, feparated, and according to their
degrees of refradion, difperfed to feveral places, fo as to
conftitute fo many feveral waves, as there were divers
orders of Colours promifcuoufly relieded from that
part of the Glafs.

Thefe are the principal Phaenomena of thin Plates
©r Bubbles, whole explications depend on the pro-
perties of Light, which I have heretofore delivered;
And thefe you fee do neceflarily follow from them, and
agree with them, even to their very leaft circumftances;
and not only fo, but do very much tend to their proof.
Thus, by the a4th Obfervation, it appears, that tlie

rays

vay 6 oT feveidl Colours made as well by thin Plates or
Bubbles, as by refractions of a Prifm, have leveral de-
grees of refrangibility, whereby thole of each order,
which at their reflexion from the Plate or Bubble are
intermixed with thofe of other orders, are feparated
•from them by refraction, and aflbciated together lb as to
.become vifibleby themlelves like Arcs ot Circles. For
if the rays were all alike refrangible, 'tis impoffible that
the whiteneis, which to the naked fence appears uni-
form, iliould by refraction have its parts tranipoied and
.ranged into thofe black and white Arcs.

It appears alio that the unequal refraClions of dif-
form rays proceed not from any contingent irregulari-
ties ; fuch as are veins, an uneven polifh, or fortuitous
portion of the pores of Glafs ; unequal and cafual mo-
tions in the Air or ^ther ; the fpreading, breaking, or
dividing the fame ray into many diverging parts, or
the like. For, admitting any fuch irregularities, it would
he impoflible for refractions to render thofe Rings {o
very diftinCt , and well defined , as they do in the
a^-th Obfervation. It is neceflary therefore that eve-
ry ray have its proper and conftant degree of refran-
gibility connate with it,according to which its refraCtion
is ever julHy and regularly performed, and that feve-
ral rays have leveral of thole degrees.

And what is laid of their refrangibility may be alfo
underftood of theii* refiexibility, that is of their difpo-
fitions to be reflected fome at a greater, and others at a
lefs thicknefs, of thin Plates or Bubbles, namely, that
thofe difpofltions are alfo connate with the rays, and
immutable j as may appear by the i^th, i^-th, and

15th

[47] .

»5th Obfervations coinpared with the fourth and
eighth.

By the precedent Obfervations it appears aUb, that
whitenefs is a diffimilar mixture of all Colours, and that
Light is a mixture of rays indued with all thofe Co-
lours. For conlidering the multitude of the Rings of
Colours, in the ^d, 1 2th and 14-th Obfervations, it is
manifefl: that although in the 4th and 1 8th Obferva-
tions there appear no more than eight or nine of thofe
Rings, yet there are really a far greater number, which
fo much interfere and mingle with one another, as after
thofe eight or nine revolutions to dilute one another
wholly.^ and conlHtute an even and fenfibly uniform
whitenefs. And confequently that whitenefs muft be
allowed a mixture of all Colours, and the Light which
conveys it to the Eye muft be a mixture of rays indued,
with ail thofe Colours.

But further, by the a^th Obfervation , it appears,
that there is a conftant relation between Colours and
Refrangibility, the moft refrangible rays being violet,
the leaft: refrangible red, and thofe of intermediate Co-
lours having proportionably intermediate degrees of re*
frangibility. And by the 13 th, 14th and 15 th Obfer-
vations, compared w^ith the 4th or 1 8th, there appears
to be the fame conftant relation between Colour and
Reflexibility, the violet being in like circumftances re^
Hefted at leaft thickneffes of any thin Plate or Bubble,
the red at greateft thickneffes , and the intermediate
Colours at intermediate thickneffes. Whence it fol-
lows, that the colorifique difpofitions of rays are alfo
connate with them and immutable, and by confequence

that

[48]

that all the produ(5tions and appearances of Colours
in the World are derived not from any phyfical change
caufed in Light by refraction or reflexion, but only
from the various mixtures or feparations of rays, by
virtue of their diflferent Refrangibility or Reflexibility.
And in this refped: the Science of Colours becomes a
Speculation as truly mathematical as any other part of
Optiques. 1 mean fo far as they depend on the nature
of Light, and are not produced or altered by the power
of imagination, or by ftriking or prefling the Eyes.

THE

C49]

THE

SECOND BOOK

O F

O P T I C K S.

PART III.

Of the permanent Colours of natural Bodies^ and the
u4nalogy hetisjeen them and the Colours of thin tranf*'
parent Tlates.

I Am now come to another part of this Defign, which
is to confider how the Phsenomena of thin tranfpa-
rent Plates (land related to thofe of all other natural
Bodies. Of thefe Bodies I have already told you that
they appear of divers Colours, accordingly as they are
difpofed to refledt moft copioufly the rays originally
indued with thofe Colours. But their Conftitutions,
whereby they reflect fome rays more copioufly than
others, remains to be difcovered, and thefe I fliall en-
deavour to manifeft in the following Propofitions.

Hh PROP,

C50]

PROP. I.

T^hojefuperjiciesoftrcinfparent Bodies rejledl thegreatejf
quantity of Light y'which have the greate^ref racing foisoey,
that is^ isjhich intercede mediums that differ mojl in their
refraHive denfities. ^nd in the confines of equoMy re-
fraHing mediums there js no reflexion.

The Analogy between reflexion and refradion will
appear by conlidering, that when Light palTeth ob-
hquely out of one medium into another which refrad:?
from the perpendicular, the greater is ditference of
their refractive denfity, the lets obliquity is requifite
to caufe a total reflexion. For as the Sines are which
meafure the refraction, fo is the Sine of incidence at
which the total reflexion begins, to the radius of the
Circle, and confequently that incidence is leaft where
there is the greatelt ditference of the Sines. Thus in the
pafling of Light out of Water into Air, where the
refraction is meafured by the Ratio of the Sines g to 4.,
the total reflexion begins when the Angle of incidence
is about 48 degrees 55 minutes. In pafling out ofGlafs
into Air, where the refraCtion is meafured by the Ratio
of the Sines 20 to gi, the total reflexion begins when.
the Angle of incidence is 40 deg. 10 min. and fo irt
paffing out of cryftal, or more ftrongly refraCting me-
diums into Air, there is ftill a lefs obliquity requifite
to caufe a total i^eflexion. Superficies therefore which
refraCt mofl: do fooneft refleCt all the Light which is in^
cident on them, and fo muft be allowed moft fl:rongly
reflexive.

But

[51].

But the truth of this Propofition will further appear
by obferving , that in the luperficies interceding two
tranfparent mediums, fuch as are ( Air, Water ,Oyl, Com'
mon-Glafs, Cryftal, MetaUine-GlafTes^ Ifland-GlafTesj
white tranfparent Arfnick, Diamonds, ]5^c. ) the re-
flexion is ftronger or weaker accordingly, as the fuper-
ficies hath a greater or lefs refracting power. For in
the confine of Air and Sal-gemm 'tis ftronger than in
the confine of Air and Water, and iHll ftronger in the
confine of Air and Common-Glafsor Cryftal,and ftronger
in the confine of Air and a Diamond. If any of thefe,and
fuch like tranfparent Solids, be immerged in Water, its
reflexion becomes much weaker than before, and ftill
weaker if they be immerged in the more ftrongly re-
framing Liquors of well-redified oyl of Vitriol or fpirit
of Turpentine. If Water be diftinguifhed into two parts,
by any imaginary furface, the reflexion in the confine
of thofe two parts is none at all. In the confine of Wa-
ter and Ice 'tis very little, in that of Water and Oyl 'tis
fomething greater, in that of Water and Sal-gemm ftill
greater, and in that of Water and Glafs, or Cryftal, dr
other denfer fubftances ftill greater, accordingly as thofe
mediums ditfer more or lefs in their refrading powersi.
Hence in the confine of Common-Glafs and Cryftal,
there ought to be a weak reflexion, and a ftronger re-
flexion in the confine of Common and Metalline-Glafs,
though I have not yet tried this. But, in the confine of
two Glafles of equal denfity, there is not any lenfible re-
flexion, as was ihewn in the firft Obfervation, ,And
the fame may be underftood of the fuperficies iittercer
ding two Cryftals, or two Liquors, or any other Subr
ftances in which no refradion is caufed. So theri tte

Hh '2 reafon

C50

reafon why uniform pellucid mediums, (fuch as Water,
Glafs, or Cryftal) have no feniible reflexion but in
their external fuperficies, where they are adjacent to
other mediums of a different denfity , is becaufe all
their contiguous parts have one and the fame degree
of denfity.

PROP. II.

T'he leajl farts of almofl all natural Bodies are in fome
meafure tranffarent : jind the opacity of thoje Bodies
arijeth from the multitude of reflexions caufed in their in^
ternal ^arts.

That this is fo has been obferved by others, and
will eafily be granted by them that have been conver-
fant with Mifcrofcopes. And it may be alio tryed by
applying any fubftance to a Hole through which fome
Light is immitted into a dark room. For how opake
foever that fubftance may feem in the open Air, it wili
by that means appear very manifeftly tranfparent, if
it be of a fufticient thinnefs. Only white metalline Bo-
dies muft be excepted, which by reafon of their excef-
five denfity feem to relied: almoft all the Light inci-
dent on their firft fuperficies , unlefs by folution in
menftruums they be reduced into very fmall particlesj^
and then they become tranfparent.

PROP. III.

Betisueen the farts of ofahe and coloured Bodies are
manyfpaces^ either emfty or reflenijhed^ isuith mediums
of other denftties ; as JVater l^etisjeen the tinging corfufcles
"wherewith any Uquor is impregnated^ jiir bet'ween the

aqueous

f^.53 3

aqueom glohules that confiitute Clouds or Mifts ; and for
the moji fart [faces void of both u4ir and Water ^ hut yet
ferhafs not 'wholly void of all fuSJiance^ between the farts
of hard Bodies.

The truth of this is evinced by the two precedent
Propofitions : For by the fecond Proportion there are
many reflexions made by the internal parts of Bodies,
which, by the firft Propolition, would not happen if
the parts of thofe Bodies were continued without any
fuch interftices between them, becaufe reflexions are
caufed only in fuperficies, which intercede mediums of
a differing density by Prop, i .

But further, that this difcontinuity of parts is the
principal cauie of the opacity of Bodies, will appear by
confidering, that opake fubftances become, tranfparent
by filling their pores with any fubfliance of equal or al-
moft equal denfity with their parts. Thus Paper dip-
ped in Water or Oyl, the Oculm mundi Stone fteep'd in
Water, Linnen-cloth oyled or varnifhed, and many other
fubftances foaked in fuch Liquors as will intimately
pervade their little pores, become by that means more
tranfparent than otherwife -, fo, on the contrary, the
moft tranfparent fubftances may by evacuating their
pores, or leparating their parts, be rendred fufficiently
opake, as Salts or wet Paper, or the Oculm mundt Stone
by being dried, Horn by being fcraped, Glafs by being
reduced to powder, or otherwife flawed, Turpen-
tine by being ftirred about with Water till they mix
imperfectly , and Water by being formed into many
fmall Bubbles, either alone in the form of froth, or
by fliaking it together with Oyl of Turpentine, or
with fome other convenient Liquor, with which it will

not

[54]

not peitedly incorporate. And to the increafe of the
opacity of thefe Bodies it conduces fomething, that by
the a^thObfervation the reflexions of very thin tranf-
parent fubftanccs are conliderably ilronger than thofe
made by the fame fubfl:ances of a greater thicknefs.

PROP. IV.

T^he farts of Bodies and their Inter flues muji not be
lefs than offome definite hignejs^ to render them opake and
< coloured.

For the opakeft Bodies, if their parts be fubtily
divided, ( as Metals by being diflblved in acid men-
llruums, }^c.) become perfedly tranfparent. And you
may aUb remember, that in the eighth Obfervation
there was no fenlible reflexion at the fuperficies of
the Obje£t-Glafles u^here they w^ere very near one
another, though they did nc^ abfolutely touch. And
in the 1 7 th Obfervation the reflexion of theWater-bubble
where it became thinneft was almoft infenflble, fo as
to caufe very black Spots to appear on the top of the
Bubble by the want of refleded Light.

On thefe grounds I perceive it is that Water, Salt,
Glafs, Stones, and fuch like fubfl^nces, are tranfparent.
For, upon divers coniiderations, they feem to be as full
of pores or interftices between their parts as other Bo-
dies are, but yet their parts and interflices to be too
fmali to caufe reflexions in their common furfaces.

PROP

[$5]

PROP. V.

T'he tranffarent farts of Bodies according to their fe-
veral fizes muji reJle<H rays of one Colour^ and tranfmtt
thofe of another J on the fame grounds that thtn "Plates or
BuSMes do rejleSl or tranfmtt thofe rays. And this J take
to be the s^rotmd of all their Colours.

For if a thin'd or plated Body, which being of an
even thicknefs, appears all over of one uniform Co-
lour, fhould be ilit into threds, or broken into frag-
ments, of the fame thicknefs with the plate ; I fee no
reafon why every thred or fragment fhould not keep its
Colour, and by confequence why a heap of thofe threds
or fragments fhould not conftitute a mafs or powder of
the fame Colour, which the plate exhibited before it
was broken. And the parts of all natural Bodies being
like fo many fragments of a Plate, muft on the fame
grounds exhibit the fame Colours.

Now that they do fo, will appear by the afhniiy of
their properties. The finely coloured Feathers of fome
Birds, and particularly thofe of Peacocks Tails, do in
the very fame part of the Feather appear of feveral Co-
lours in feveral pofitions of the Eye, after the very fame
manner that thin Plates were found to do in the 7th
and 19 th Obfervations , and therefore arife from the
thinnefs of the tranfparent parts of the Feathers ; that
is, from the flendernefs of the very fine Hairs, or Cafilla-
menta^ which grow out of the fides of the grofler late-
ral branches or fibres of thofe Feathers. And to the
lame purpofe it is, that the Webs of fome Spiders by

being

being fpun very fine have appeared coloured, as Ibme
have obierved, and that the coloured fibres of ibme filks
by varying the pofition of the Eye do vary their Co-
lour. AUb the Colours of filks, cloths, and other fub-
ftances, which Water or Oyl can intimately penetrate,
become more faint and obfcure by being immerged in
thofe liquors, and recover their vigor again by being
dried, much after the manner declared of thin Bodies
in the loth and "^ith Obfervations. Leaf-gold, fome
forts of painted Glafs, the infufion of Lignum Mefhri-
ticum^ and fome other fubftances reflect one Colour,
and tranfmit another, like thin Bodies in the 9th and
aoth Obfervations. And fome of thofe coloured pow-
ders which Painters ufe, may have their Colours a little
changed, by being very elaborately and finely ground.
Where 1 fee not what can be juftly pretended for thofe
changes, befides the breaking of their parts into lefs
parts by that contrition after the fame manner that the
Colour of a thin Plate is changed by varying its thick-
nefs. For which reafon alfo it is that the coloured flowers
of Plants and Vegitables by being bruifed ufually be-
come more tranfparent than before, or at leaft in fome
degree or other change their Colours. Nor is it much
lefs to my purpofe, that by mixing divers liquors very
odd and remarquable produdions and changes of Co-
lours may be effected, of which no caufe can be more
obvious and rational than that the faline corpufcles of
one liquor do varioufly aft upon or unite with the
tinging corpufcles of another, fo as to make them fwell,
or Ihrink (whereby not only their bulk but their den-
iity alfo may be changed ) or to divide them into
fraaller corpufcles, (whereby a coloured liquor may be-
come

D 57 ]

come tranfparcnt) or to make many of them aflbclate
into one clufter, whereby two tranfparent liquors may^
compoie a coloured one. For we fee how apt thofe
faline menftruums are to penetrate and diflfolve fuh-*
ftances to which they are applied, and fome of them
to precipitate what others diffolve. In like manner, if
we conlider the various Phsenomena of the Atmofphasre,;
we may obferve, that when Vapors are firft raifed, they
hinder not the tranfparency of the Air, being divided
into parts too fmall to caufe any reflexion in their fuper-;
ficies. But when in order to compofe drops of rain they
begin to coalefce and conftitute globules of all interi.:
mediate fizes, thofe globules when they becorhe of a^
convenient fize to reflect fome Colours and tranfmit
others, may conftitute Clouds of various Colours accor*.
ding to their hzes. And I fee not what can be ratio-^
nally conceived in fo tranfparent a fubftance as Water for
the produdion of thefe Colours, befides the various
fizes of its fluid and globuler parcels. '

PROP. VI.

The farts of Bodies on which their Colours depend^
are denjer than the medium , which pervades their in*
terjlices.

This will appear by confldering, that the Colour of
a Body depends not only on the rays which are inci-
dent perpendicularly on its parts, but on thofe alfo
which are incident at all other Angles. And that ac*
cording to the yth Obfervation, a very little variation
of obliquity will change the reflected Colour where the
thin body or fmall particle is rarer than the ambient,

I i medium,

^58]

medium, infomuch that fuch a fmall particle will at di-
verily oblique incidences relied all forts of Colours, in
fo great a variety that the Colour rel'ulting from them
all, confufedly reflected from a heap of fuch particles,
muft rather be a white or grey than any other Colour,
or at beft it muft be but a very imperfedl and dirty Co-
lour. Whereas if the thin body or fmall particle be
much denfer than the ambient medium, the Colours
according to the 19th Obfervation are fo little changed
by the variation of obliquity, that the rays which are
reflected leaft obliquely may predominate over the reft
fo much as to caufe a heap of fuch particles to appear
very intenfly of their Colour.

It conduces alfo fomething to the confirmation of this
Propofition, that, according to the a 2th Obfervation,
the Colours exhibited by the denfer thin body within
the rarer, are more brifque than thofe exhibited by the
rarer within the denter.

PROP. VII.

The hignejs of the component farts of natural Bodies
may be con^eBured by their Colours.

For lince the parts of thefe Bodies by Prop. 5. do
moft probably exhibit the fame Colours with a Plate of
equal thicknefs, provided they have the fame refractive
denfity ; and fince their parts feem for the moft part to
have much the fame deniity with Water or Glafs, as
by many circumftances is obvious to colled ; to deter^
mine the lizes of thofe parts you need only have recourfe
to the precedent Tables, in which the thicknefs of Wa-
ter or Glafs exhibiting any Colour is expreffed. Thus

[59]

if it be defired to know the Diameter of a corpufcle,
which being of equal denfity with Giafs fhall refled:
green of the third order ; the number 1 6^ fhews it to
be "^4 parts of an Inch.

lOOOOO

The greateft difficulty is here to know of what order
the Colour of any Body is. And for this end we muft
liave recourfe to the 4.th and 1 8th Obfervations, from
whence may be collected thefe particulars.

Scarlets^ and other I'eds^ oranges and yeSo'ws^ if they
be pure and intenfe are moft probably of the fecond or-
der. Thole of the firft and third order alfo may be
pretty good, only the yellow of the firft order is faint,
and the orange and red of the third order have a great
mixture of violet and blue.

There may be good greens of the fourth order, but
the pureft are of the third. And of this order the green
of all vegitables feem to be, partly by reafon of the in-
tenfenefs of their Colours , and partly becaufe when
they wither fome of them turn to a greenifh yellow;,
and others to a more perfect yellow or orange, or per-
haps to red, paffing firft through all the aforefaid in-
termediate Colours. Which changes feem to be effected
by the exhaling of the moifture which may leave the
tinging corpufcles more denie, and fomething augmen-
ted by the accretion of the oyly and earthy part of
that moifture. Now the green without doubt is of the
fame order with thofe Colours into which it changeth,
becaufe the changes are gradual, and thofe Colours,
though ufually not very full, yet are often too full and
lively to be of the fourth order.

I i 2 Blues

[6o]

Blue's and ptrfles maybe either of the fecond or third
order, but the beft are of the third. Thus the Colour
of violets feems to be of that order, becaufe their Syrup
by acid Liquors turns red, and by urinous and alcali-
zale turns green. For lince it is of the nature of Acids
<to diflblve or attenuate, and of Alcalies to precipitate
or incralTate, if.the purple Colour of the Syrup was of
the fecond order, an acid Liquor by attenuating its ting-
ing corpufcles would change it to a red of the firft
order, and an Alcaly by incralTating them would change
it to a green of the fecond order ; which red and green,
efpecially the green, feem too imperfed to be the Co-
lours produced by thefe changes. But if the faid purple
be fuppofed of the third order, its change to red of the
Second, and green of the third, may without any in-
convenience be allowed. . ,

, , If there be found any Body of a deeper and lefs redr
.difh purple than that of the violets, its Colour moft
probably is of the fecond order. But yet their being
no Body commonly known whofe Colour is conftantly
^more deep than theirs, I have made ufe of their name to
(denote the deepeft and leaft reddilb piuples, fuch as
manifeftly tranicend their Colour in purity.

The Mue of the firit order , though very faint and
little, may poffibly be the Colour of fome fubftances ;
and particularly the azure Colour of the Skys feems to
be ot this order. For all vapours when they begin to
condenfe and coalefce into fmall parcels, become tirft of
that bignefs whereby fuch an Azure muft be retle6led
before they can conftitute Clouds of other Colours. And
io this being the firft Colour which vapors begin to
xefledl, it ought to be the Colour of the hneft and moft

traiif-

i6i}

tranfparent Skys in which vapors are not arrived to that
grolhefs requifite to refled other Colours^ as we find it
is by experience.

JVhitenefs^ if moft intenfe and luminous, is that of the
fir ft order, if lefs ftrong and luminous a mixture of the
Colours of feveral orders. Of this laft kind is the
whitenefs of Froth, Paper, Linnen, and moft white fub-
ftances 3 of the former I reckon that of white metals to
be. For whilft the denfeft of metals, Gold, if foliated
is tranfparent, and all metals become tranfparent if
diflblved in menftruums or vitrified, the opacity of
white metals arifeth not from their denfity alone. They
being lefs denfe than Gold would be more tranfparent
than it, did not fome other caufe concur with their den^
:fity to make them opake. And this caufe I take to be
fuch a bignefs of their particles as. fits them to refled
the white of the firft order. For if they be of other
thicknelTes they may reflect other Colours, as is mani-
feft by the Colours which appear upon hot Steel in tem-
pering it, and fometimes upon the furface of melted
metals in the Skin or Scoria which arifes upon them in
their cooling. And as the white of the firft order is
the ftrongeft which can be made by Plates of tranfparent
fubftances, fo it ought to be ftronger in the denfer fub-
ftances of metals tlian in the rarer of Air, Water and
Glafs. Nor do 1 fee but that metallic fubftances of fuch
a thicknefs as may fit them to reflect the white of the
firft order, may, by reafon of their great denfity (accor-
ding to the tenour of the firft of thefe Propofitions) re*^
tied all the Light incident upon them, and fo be as
opake and fplendent as its poffible for any Body to be.
Gold, or Copper mixed with lefs than half their weight

of Silver, or Tin, or Regulus of Antimony, in fufion
or amalgamed with a very little Mercury become white;
which ihews both that the particles of white metals
have much more fuperftcies, and fo are fm/aller, than
thofe of Gold and Copper, and alfo that they are lb
opake as not to fuffer the particles of Gold or Copper to
fhine through them. Now it is fcarce to be doubted,
but that the Colours of Gold and Copper are of the fe-
cond or third order, and therefore the particles of white
metals cannot be much bigger than is requifite to make
them reflect the white of the firft order. The volati-
lity of Mercury argues that they are not much bigger,
nor may they be much lefs, leaft they lofe their opacity,
and become either tranfparent as they do when attenua-
ted by vitrification, or by folution in menftruums, or
black as they do when ground fmaller, by rubbing Sil-
ver,or Tin, or Lead, upon other fubftances to draw black
Lines. The firtt and only Colour which white metals
take by grinding their particles fmaller is black, and
therefore their white ought to be that which borders
upon the black Spot in the center of the Rings of Co-
lours, that is, the white of the lirft order. But if you
would hence gather the bignefs of metallic particles,
you muft allow for their denlity. For were Mercury
tranfparent, its denlity is fuch that the Sine of inci-
dence upon it (by my computation) would be to the
fine of its refraction, as 71 to ao, or 7 to a. And
therefore the thicknels of its particles, that they may
exhibit the fame Colours with thofe of Bubbles of Wa-
ter, ought to be lefs than the thicknels of the Skin of
thofe Bubbles in the proportion of a to 7. Whence
its poffible that the particles of Mercury may be as little

as the particles of Tome tranfparent and volatile fluids,
and yet relied the white of the firft order.

Laftly, for the produ<Sion o( Mach^ the corpufcies
muft be lefs than any of thofe which exhibit Colours.
For at all greater fizes there is too much Light refle-
ded to conlHtute this Colour. But if they be fuppo-
fed a little lefs than is requiiite to reflect the white and
very faint blue of the firft order, they will, according
to the 4.th, 8th, 17th and 1 8th Obfervations, refled
fo very little as to appear intenfly black, and yet may
perhaps varioully refrad: it to and fro within them-
felves fo long, until it happen to be ftifled and loft,
by which means they will appear black in all politions
of the Eye without any tranfparency. And from hence
may be underftood why Fire , and the more fubtile
diffolver Putrefadion, by dividing the particles of fub-
ftances, turn them to black , why fmall quantities of
black fubflances impart their Colour very freely and in-=
tenfly to other fubftances to which they are applied ;
the minute particles of thefc, by reafon of tkeir very
great number, ealily overfpreading the grofs particles
of others ; why Glafs ground very elaborately with
Sand on a copper Plate, 'till it be well polifhed, makes
the Sand, together with what is worn oft from the Glafs
and Copper, become very black : why black fubftances
do fooneft of all others become hot in the Sun's Light
and burn, (which effed may proceed partly from the
multitude of refradions in a little room, and partly
from the eafy commotion of fo very fmall corpufcies;)
and why blacks are ufually a little inclined to a bluiih
Colour. For that they are fo may be feen by illumina^
ting white Paper by Light refte6ted from black fub^

ftanceso

E 64 ],

ftances. For the Paper will ulually appear of a bluifli
white ; and the reafon is, that black borders on the
obfcure blue of the firft order defcribed in the i8th
Obfervation, and therefore refle6ls more rays of that
Colour than of any other.

In thefe Defcriptions I have been the more particu-
lar, becaufe it is not impoffible but that Mifcrofcopes
may at length be improved to the difcovery of the
particles of Bodies on which their Colours depend, if
they are not already in fome meafure arrived to that de-
gree of perfe(5tion. For if thofe Inftruments are or can
be fo far improved as with fufficient diftindnefs to re-
prefent Objects five or fix hundred times bigger than
at a Foot diftance they appear to our naked Eyes, I
Ihould hope that we might be able to difcovcr fome of
the greateft of thofe corpufcles. And by one that would
magnify three or four thoufand times perhaps they
might all be difcovered, but thofe which produce black-
nefs. In the mean while I fee nothing material in this
Difcourfe that may rationally be doubted of excepting
this Fofition, That tranfparent corpufcles of the fame
thicknefs and denfity with a Plate, do exhibit the fiune
Colour. And this I would have underftood not with-
out fome latitude, as well becaufe thole corpufcles may
be of irregular Figures, and many rays muft be oblique-
ly incident on them, and fo have a Ihorter way through
tiiem than the length of their Diameters, as becaufe the
ftraitnefs of the medium pent in on all fides within fuch
corpufcles may a little alter its motions or other qua-
lities on which the reflexion depends. But yet I can-
not much fufpe^t the laft, becaufe 1 have obferved of
irnn^ Ijmall Plates of Mufcovy-Glafs which were of an

L<55]

even thicknefs, that through a Mifcrofcope they have
appeared of the fame Colour at their edges and cor-
ners where the included medium was terminated, which
they appeared of in other places. However it will add
much to our fatisfadion, if thofe corpufcles could be dif-
covered with Mifcrofcopes ; which if we Ihall at length
attain to, I fear it will be the utmoft improvement of
this fenfe. For it feems impoffible to fee the more fe-
cret and noble works of nature within the corpufcles
by reafon of their tranfparency.

PROP. VIII.

T'he caufe of Reflexion is not the imfinging of Light on
the folid or im^erviom ^arts of Bodies^ m is commonly Re-
lieved.

This will appear by the following Confiderations.
Firft, That in the paflage of Light out of Glafs into
Air there is a reflexion as ftrong as in its paflage out of
Air into Glafs, or rather a little ftronger, and by many
degrees ftronger than in its paflage out of Glafs into
Water. And it feems not probable that Air fliould have
more refieding parts than Water or Glafs. But if that
fliould poflibly be fuppofed, yet it will avail nothing ;
for the reflexion is as ftrong or ftronger when the Air is
drawn away from the Glafs, (fuppole in the Air-pump
invented by Mr. Boyle ) as when it is adjacent to it.
Secondly, If Light in its paflage out of Glafs into Air
be incident more obliquely than at an Angle of 4.0 or
4.1 degrees it is wholly refleded, if lefs obliquely it is
in great meafure tranfmitted. Now it is not to be ima-
gined that Light at one degree of obliquity ftiould meet

K k with

with pores enough in the Air to tranfmit the greater
part of it, and at another degree of obliquity fhould
meet with nothing but parts to relied it wholly, efpe^
cially conhdering that in its paflage out of Air into
Glals , how oblique foever be its incidence , it finds
pores enough in the Glafs to tranfmit the greateft part
of it. If any Man fuppofe that it is not refleded by the
Air, but by the outmoft fuperficial parts of the Glafs,
there is ftill the fame difficulty : Befides, that fuch a
Suppofition is unintelligible, and will alfo appear to be
falfe by applying Water behind fome part of the Glafs
inftead of Air. For fo in a convenient obliquity of the
rays fuppofe of 45 or 4.6 degrees, at which they are all
fenefted where the Air is adjacent to the Glafs, they
fhall be in great meafure tranfmitted where the Water
is adjacent to it ; which argues, that their reflexion
or tranfmiffion depends on the conftitution of the Air
and Water behind the Glafs, and not on the ftriking
off the rays upon the parts of the Glafs. Thirdly, If
the Colours made by a Prifm placed at the entrance of
a beam of Light into a darkened room be fucceflively
caft on a fecond Prifm placed at a greater diftance from
the former, in fuch manner that they are all alike inci-
dent upon it, the fecond Prifm may be fo inclined to
the incident rays, that thofe which are of a blue Colour
fhall be all refle6ted by it, and yet thofe of a red Colour
pretty copioufly tranfmitted. Now if the reflexion be
caufed by the parts of Air or Glafs, I would ask, why
at the fame obliquity of incidence the blue fliould whol-
ly impinge on thofe parts fo as to be all reflected, and
yet the red find pores enough to be in great meafure
tranfmitted. Fourthly, where two Glafles touch one

another,

C«57]

another, there is no fenfible reflexion as was declared
in the fir ft Obfervation ; and yet I fee no reafon why
the rays fhould not impinge on the parts of Glafs as
much when contiguous to other Glafs as when con-
tiguous to Air. Fifthly, When the top of a Water-
bubble (in the 1 7th Obfervation) by the continual fub-
liding and exhaling of the Water grew very thin, there
was fuch a little and almoft infenfible quantity of Light
refleded from it, that it appeared intenlly black ; where-*
as round about that black Spot, where the Water was
thicker, the reflexion was lb ftrong as to make the
Water feem very white. Nor is it only at the leafl:
thicknefs of thin Plates or Bubbles, that there is no
manifeft reflexion, but at many other thicknefles con-
tinually greater and greater. For in the 1 5 th Obfer-
vation the rays of the fame Colour were by turns tranf-
mitted at one thicknefs, and reflected at another thxcle-
nefs, for an indeterminate number of fucceflions. And
yet in the fuperficies of the thinned Body, where it is
of any one thicknefs, there are as many parts for the
rays to impinge on, as where it is of any other thick-
nefs. Sixthly, If reflexion were caufed by the parts of
refledling Bodies, it would be impoflible for thin Plates
or Bubbles at the fame place to reflect the rays of one
Colour and tranfmit thofe of another, as they do accor-
ding to the 13 th and 15 th Obfervations. For it is
not to be imagined that at one place the rays which
for inftance exhibit a blue Colour, fliould have the for-
tune to dafli upon the parts, and thofe which exhibit
a red to hit upon the pores of the Body ; and then at
another place, where the Body is either a little thicker,
or a little thinner, that on the contrary the blue fliould

Kk 2 hit

[68]

hit upon its pores, and the red upon its parts. Laftly,
were the rays of Light refleded by impinging on the
folid parts of Bodies, their reflexions from polifhed Bo-
dies could not be fo regular as they are. For in po-
lifhing Glafs with Sand, Putty or Tripoly, it is not to
be imagined that thole fubftances can by grating and
fretting the Glais bring all its leaft particles to an ac-
curate polifh ; fo that all their furfaces fhall be truly
plain or truly fpherical, and look all the fame way, lb
as together to compofe one even furface. The fmaller
the particles of thole fubftances are, the fmaller will
be the fcratches by which they continually fret and wear
away the Glafs until it be polifhed, but be they never
fo fmall they can wear away the Glafs no otherwife
than by grating and fcratching it , and breaking the
proturberances , and therefore polifh it no otherwile
than by bringing its roughnefs to a very fine Grain, fo
that the fcratches and frettings of the furface become
too fmall to be vilible. And therefore if Light were
reflected by impinging upon the folid parts of the Glafs,
it would be fcattered as much by the moft polifhed
Glafs as by the rougheft. So then it remains a Pro^
blem, how Glafs polifhed by fretting fubftances can re-
flect Light fo regularly as it does. And this Problem
is fcarce otherwife to be folved than by faying, that
the reflexion of a ray is effected, not by a Angle point of
the reflecting Body, but by fome power of the Body
which is evenly diffufed all over its furface, and by
which it a6ts upon the ray without immediate contadt.
For that the parts of Bodies do a6t upon Light at a di-
ftance fhall be ftiewn hereafter^

Now

Now if Light be refleded not by impinging on the
folid parts of Bodies, but by fome other principle ; its
probable that as many of its rays as impinge on the
folid parts of Bodies are not refleded but ftifled and
loft in the Bodies. For othcrwife we muft allow two
forts of reflexions. Should all the rays be reflected which
impinge on the internal parts of clear Water or Cryftal,
thofe fubftances would rather have a cloudy Colour
than a clear tranfparency. To make Bodies look black,
its neceflary that many rays be ftopt, retained and loft
in them, and it feems not probable that any rays can
be ftopt and ftifled in them which do not im.pinge on
their parts.

And hence we may underftand that Bodies are much
more rare and porous than is commonly believed. Wa-
ter is 19 times lighter, and by confequence 19 times
rarer than Gold , and Gold is fo rare as very readily
and without the leaft oppofition to tranfmit the mag-
netick Effluvia, and eaftly to admit Quick-ftlver into
its pores, and to let Water pals through it. For a con-
cave Sphere of Gold filled with Water, and fodered up,
has upon prefling the Sphere with great force, let the
Water fqueeze through it, and ftand all over its out-
lide in multitudes of fmall Drops, like dew, without
burfting or cracking the Body of the Gold as I have
been informed by an Eye-witnefs. From all which we
may conclude, that Gold has more pores than folid
parts, and by confequence that Water has above forty-
times more pores than parts. And he that fliall find out
anHypothefis, by which Water may be fo rare, and yet
not be capable of compreflion by force, may doubtlefs
by the fume Hypothefis make Gold and Water, and all

othtr

[70]

Other Bodies as much rarer as he pleafes, fo that Light
may find a ready paffage through tranlpareiit fub-

ftances.

PROP. IX.

Bodies rejleH and refraB Light b>j one and the fame
fo%^er varioujly exercifed in vartom circumjiances.

This appears by leveral Confiderations. Firft, Be^
caufe when Light goes out of Glafs into Air, as ob-
liquely as it can poffibly do, if its incidence be made
ftill more oblique, it becomes totally reflected. For
the power of the Glafs after it has refraded the Light
as obliquely as is poffible if the incidence be ftill made
more oblique, becomes too ftrong to let any of its rays
go through, and by confequence caufes total reflexions.
Secondly , Becaufe Light is alternately refleded and
tranfmittcd by thin Plates of Glafs for many fucceffions
accordingly , as the thicknefs of the Plate increafes
in an arithmetical Progreffion. For here the thicknefs
of the Glafs determines whether that power by which
Glafs ads upon Light fhall caufe it to be refleded, or
fufFer it to be tranfmitted. And, Thirdly, becaufe thofe
furfaces of tranfparent Bodies which have the greateft
refrading power, refled the greateft quantity of Light,
as was fhewed in the firft Propofition.

PROP. X.

If Light be fisjifter in Bodies than in T^acuo in the
frofcrtion, of the Smes "which meajure the refracHion of the
Bodies J the forces of the Bodies to reflet and refraB Light ^

are

[71]

are very nearly proportional to the den/ities of the fame
Bodies^ excepting that unHuous and fulphureom Bodies re-
fraH more than others of this fame denfity.

Let A B rcprefent the refrading plane furface of any
Body, and I C a ray incident very obliquely upon the

Body in C, fo that the Angle A CI may be infinitely
little, and let CR be the refracted ray. From a given
point B perpendicular to the refra(5ting furface ere£i:
B R meeting with the refracted ray C R in R, and if
CR reprefent the motion of the refracted ray, and this
motion be diftinguifhed into two motions C B and B R,
whereof CB is a parallel to the refracting plane, and
BR perpendicular to it : CB (hall reprefent the motion
of the incident ray, and B R the motion generated by
the refraction, as Opticians have of late explained.

Now if any body or thing in moving through any
fpace of a giving breadth terminated on both (ides by
two parallel plains, be urged forward in all parts of
that fpace by forces tending direCtly forwards towards
the laft plain, and before its incidence on the firil
plane, had no motion towards it, or but an infinitly
little one ; and if the forces in all parts of that fpace,
between the planes be at equal diftances from the planes
equal to one another, but at feveral diftances be bigger
or lefs in any given proportion, the motion generated
by the forces in the whole paffage of the body or thing

through

[72]

through that fpace Ihall be in a fubduplicate proportion
of the forces, as Mathematicians will eafily underftand.
And therefore if the fpace of activity of the refracting
fuperficies of the Body be confidered as fuch a fpace,
the motion of the ray generated by the refrading force
of the Body , during its paflage through that fpace
that is the motion BR muft be in a fubduplicate
proportion of that refracting force : I fay therefore that
the fquare of the Line B R, and by confequence the
refracting force of the Body is very nearly as the den-
fity of the fame Body. For this will appear by the fol-
lowingTable, wherein the proportion of the Sines which
meafurc the refraxions of feveral Bodies, the fquare
of BR fuppofing CB an unite, the denfities of the
Bodies eftimated by their fpecifick gravities, and their
refradive power in refped of their denfities are fet
down in feveral Columns.

The

The refrading Bodies.

[73 3

The Proportion
of the Sins s oj
incidence and
refraction of
yellow Light.

A Pfeudo-Topazius, be-
ing a naturaljpellucid,
brittle, hairy Stone, of
a yellow Colour

Air

Glafs of Antimony

A Selenitis

Glafs vulgar

Cryftal of the Rock

Ifland Cryftal

Sal Gemma

Alume

Borax

Niter

Dantzick Vitriol

Oyl of Vitriol

Rain Water

Gumm Arabic

Spirit of Wine well re£li
fied

Camphire

Oyl Olive

Lintfeed Oyl

Spirit of Turpentine

Ambar

A Diamond

3851 to

3850

17 to

61 to

31 to

25 to

5 to

17 to

35 to

22 to

32 to

303 to

200

10 to

529 to

396

31 to

100 to

3 to

22 to

40 to

25 to

14 to

100 to

The Square of The den fit)
B R, to which and fpeci
the refracltng\ fie gravity
force oftheBoJ of the Bo-
dy is propor- dy.
ttonate.

0^00052
2'568

l'2I3

l'4025

I '44 5
i'778

i'388

1^1267

i'i5ii

i'295
i'o4i
o'7845
i'i79

©'8765

I'25

i'i5ii

i'i948
i'i626

l'42

4'949

4'27

o 00125
5'28

2'252

2'58
2^65

2'72

2'i43

i'7i4
1^714

i'9
i'7i5

i'7
1.

i'S75

o'866

©'996
0^913
o'932

o'874
I '04

3'4

The refra-
Btvepower
of the Body
in refpe£i
of its den-

3979

4160
4864
5386

54?6
5450
6536

6477
6570
6716
7079

7551
6124

7845

8574

10121

12551

12607
12819
13222
13654
14556

The refradion of the Air in this Table is determined
by that of the Atmofphere obferved by Aftronomers.
For if Light pafs through many refracting fubftances or
mediums gradually denfer and denfer, and terminated

L 1 with

[74]

with parallel furfaces, the fumm of all the refraftions
will be equal to the iingle refradtion which it would
have fuffered in palling immediately out of the firft
medium into the la ft. And this holds true, though the
number of the refracting fubftances be increafed to infi-
nity, and the diftances from one another as much de-
creafed, fo that the Light may be rcfradted in every
point of its pafTage, and by continual refradions bent
into a curve Line. And therefore the whole refraction
of Light in paffing through the Atmofphere from the
higheft and rareft part thereof down to the loweft and
denfeft part, muft be equal to the refraction which it
would futfer in paffing at like obliquity out of a Va-
cuum immediately into Air of equal deniity with that
in the loweft part of the Atmofphere.

Now, by this Table, the refraCtions of a Pfeudo-To-
paz, aSelenitis, Rock Cryftal, Ifland Cryftal, Vulgar
Glafs ( that is. Sand melted together ) and Glais of
Antimony, which are terreftrial ftony alcalizate con-
cretes,and Air which probably arifes from fuch fubftances
by fermentation,though thefe be fubftances very differing
from one another in denfity, yet they have their refra-
ctive powers almoft in the lame proportion to one ano-
ther as their denfities are, excepting that the refraCtionof
that ftrange fubftance Illand-Cryftal is a little bigger
than the reft. And particularly Air, which is 5 400 times
rarer than thePfeudo-Topaz, and 4000 times rarer than
Glafs of Antimony, has notwithftanding its rarity the
fame refraCtive power in refpeCt of its deniity which
thofe two very denfe fubftances have in refped of theirs,
excepting fo far as thofe two differ from one another.

Again,

[75]

Again, the refra£^ion of Camphire, Oyl'Olive, Lint-
feed Oyl, Spirit of Turpentine and Amber, which are
fat fulphureous unduous Bodies, and a Diamond, which
probably is an unduous fubftance coagulated, have their
refractive powers in proportion to one another as their
denfities without any confiderable variation. But the
refradive power of thefe und:uous fubftances is two
or three times greater in refped of their denfities than
the refractive powers of the former fubftances in refpeCt
of theirs.

Water has a refractive power in a middle degree be-
tween thofe two forts of fubftances, and probably is of
a middle nature. For out of it grow all vegetable and
animal fubftances, which confift as well of fulphureous
fat and inflamable parts, as of earthy lean and alcali*
zate ones.

Salts and Vitriols have refradive powers in a middle
degree between thofe of earthy fubftances and Water,
and accordingly are compofed of thofe two forts of fub*
ftances. For by diftillation and rectification of their
Spirits a great part of them goes into Water, and a great
part remains behind in the form of a dry fixt earth ca-^
pable of vitrification.

Spirit of Wine has a refraCtive power i-n a middle
degree between thofe of Water and oyly fubftances, and
accordingly feems to be compofed of both, united by
fermentation ; the Water, by means of fome faline Spi-
rits with which 'tis impregnated, diflblving the Oyl,
and volatizing it by the aCtion. For Spirit of Wine is
inflamable by means of its oyly parts, and being diftil-
led often from Salt of Tartar, grows by every diftilla-
tion more and more aqueous and flegmatick. And

LI 2 Chymifts

[7,6]

Chymlfts obierve, that Vegitables (as Lavender, Rue,
Marjoram, If^c.) diftilled fer fe , before fermentation
yield Oyls without any burning Spirits, but after fer-
mentation yield ardent Spirits without Oyls : Which
fhews, that their Oyl is by fermentation converted into
Spirit. They find alfo, that if Oyls be poured in fmall
quantity upon fermentating Vegetables, they diftil over
after fermentation in the form of Spirits.

So then, by the foregoing Table, all Bodies feemto
have their refradive powers proportional to their
denfities, ( or very nearly ; ) excepting fo fiir as they
partake more or lefs of iulphurous oyly particles, and
thereby have their refractive power made greater or
lefs. Whence it feems rational to attribute the refra-
d:ive power of all Bodies chiefly, if not wholly, to the
Iulphurous parts wdth which they abound. For it's
probable that all Bodies abound more or lefs with Sul-
phurs. And as Light congregated by a Burning-glafs
ads moft upon fulphurous Bodies, to turn them in-
to fire and flame ; fo, fince all adion is mutual, Sul-
phurs ought to ad mofl: upon Light. For that the
adion between Light and Bodies is mutual, may appear
from this Confideration, That the denfeft Bodies which
refrad and refled Light moft ftrongly grow hotteft in
the Summer-Sun, by the adion of the refraded or re-
fleded Light.

I have hitherto explained the power of Bodies to re-
fled and refrad, and Ihewed, that thin tranfparent
plates, fibres and particles do, according to their feveral
thicknefles and denfities, refled feveral ibrts of rays,
and thereby appear of feveral Colours, and by conle-
quence that nothing more is requifite for producing all

the

[77]

the Colours of natural Bodies than the feveral fizes and
denfities of their tranfparent particles. But whence it
is that thefe plates, fibres and particles do, according
to their feveral thicknefles and denfities, relied feveral
Ibrtsofrays, I have not yet explained. To give fome
infight into tiiis matter, and make way for underftan-
ding the next Part of this Book, I fhall conclude this
Part with a few more Propofitions. Thofe which pre-
ceded refped the nature of Bodies, thefe the nature of
Light : For both muft be under ftood before the reafon
of their actions upon one another can be known. And
becaufe the la ft Propofition depended upon the velo'
city of Light, I will begin with a Propofition of that
kind.

PROP. XL

Light ts frofagated from luminoim Bodies tn ttme^ and
[■pends about feven or eight minutes of an hour in faffing
from the Sun to the Earth.

This was obferved firft by Romer., and then by others^
by means of the Eclipfes of the Satellites of Jupter..
For thefe Eclipfes, when the Earth is between the Sun
and ^ufiter^ happen about feven or eight minutes fooner
than they ought to do by the Tables, and when the Earth
is beyond the Sun they happen about feven or eight mi-
nutes later than they ought to do; the reafon being, that
the Light of the Satellites has farther to go in the latter
cafe than in the former by the Diameter of the Earth's
Orbit. Some, inequalities of time may arife from the
excentricities of the Orbs of the Satellites ; but thofe
cannot anfwer in all the Satellites, and at all times

[78]

to the pofition and diftance of the Earth from the Sun.
The mean motions of Juf iter's Satellites is alfo fwifter
in his defcent from his ApheUum to his PeriheUum,
than in his afcent in the other half of his Orb : But this
inequality has no refped to the pofition of the Earth,
and in the three interior Satellites is infenfible, as I find
by computation from the Theory of their gravity.

PROP. XII.

E.ver<j 7'ay of Light in its -pajfage through any refra*
Bing jurface is fut into a certain tranfient conjiitution
or pate ^ 'which in the frogrejs of the ray returns at
equal intervals^ and dijfojes the ray at every return
to be eafdy tra?ifmitted through the next refracting fur^
face^ and between the returns to he eafdy rejie^ed by
it.

This is manifeft by the 5th, 9th, 1 ith and 1 5th Ob-
fervations. For by thofe Oblervations it appears, that
one and the fame fort of rays at equal Angles of inci-
dence on any thin tranfparent plate, is alternately refle-
cted and tranfmitted for many fucceflions accordingly,
as the thicknefs of the plate increafes in arithmetical
progreffion of the numbers o, i, a, 5,4, 5, 6, 7, 8, i^r.
fo that if the firll reflexion (that which makes the firft
or innermofl: of the Rings of Colours there defcribed )
be made at the thicknefs i,the rays fliallbe tranfmitted at
the thicknefles o, a, 4, 6, 8, 10, la, b'r. and thereby
make the central Spot and Rings of Light, which ap-
pear by tranfmiflion, and be reflefted at the thicknefs
'? 3? 5) 7) 93'^ ^^^c. and thereby make the Rings which

appear

[79]

appear by reflexion. And this alternate reflexion and
tranfmiflion, as I gather by the a^th Obfervation, con-
tinues for above an hundred viciflitudes, and by the
the Obfervations in the next part of this Book, for many
thoufands, being propagated from one furface of a Glafs-
plate to the other, though the thicknefs of the plate
be a quarter of an Inch or above : So that this alter-
nation feems to be propagated from every refrading
furface to all diftances without end or limitation.

This alternate reflexion and refradion depends on
both the furfaces of every thin plate, becaufe it de-
pends on their difl:ance. By the a i th Obfervation, if
either furface of a thin plate of Mufcovy-Glafs be wet-
ted, the Colours caufed by the alternate reflexion
and refraction grow faint, and therefore it depends on
them both.

It is therefore performed at the fecond furface, for
if it were performed at the firft:, before the rays ar-
rive at the fecond, it would not depend on the fe-
cond.

It is alfo influenced by fome adion or difpofition,
propagated from the firfl: to the fecond, becaufe other-
wife at the fecond it would not depend on the firfl:. And
this aftion or difpofition, in its propagation, intermits
and returns by equal intervals, becaufe in all its pro-
grefs it inclines the ray at one diftance from the firfl
furface to be reflefted by the fecond, at another to be
tranfmitted by it, and that by equal intervals for innu-
merable viciflitudes. And becaufe the ray is difpofed
to reflexion at the diftances i, 3, 5, y, 9, iS)'^. and to
tranfmiflion at the diftances o, a, 4., 6, 8, 10, }^c-^ ( for
its tranfmiflion through the firft furface, is at the di-
ftance

[8o]

fiance o, and it is tranfmitted through both toge-
ther, if their diftance be infinitely Httle or much lefs
than I ) the difpolition to be tranfmitted at the diftances
a, 4., 6, 8, 10, Iffc. is to be accounted a return of the
lame difpofition which the ray firft had at the diftanceo,
that is at its tranfmiffion through the firlt refracting fur-
face. All which is the thing I would prove.

What kind of adion or difpofition this is ? Whether
it confift in a circulating or a vibrating motion of the
ray, or of the medium, or Ibmething elfe ? I do not
here enquire. Thofe that are averfe from aflenting to
any new difcoveries, but fuch as they can explain by an
Hypothehs, may for the prefent fuppofe, that as Stones
by falling upon Water put the Water into an undula-
ting motion, and all Bodies by percuffion excite vibra-
tions in the Air; fo the rays of Light, by impinging on
any refracting or refledting furface, excite vibrations in
the refracting or reflecting medium or fubftance, and
by exciting them agitate the folid parts of the refraCting
or reflecting Body, and by agitating them caufe the Body
to srow warm or hot : that the vibrations thus excited
are propagated in the refraCtmg or reflecting medium
or fubftance, much after the manner that vibrations are
propagated in the Air for caufing found, and move
rafter than the rays fo as to overtake them ; and that
when any ray is in that part of the vibration which con-
fpires with its motion, it eafily breaks through a re-
traCting furface, but when it is in the contrary part of
the vibration which impedes its motion, it is eafily
reflected ; and, by confequence, that every ray is fuc-
ceffively difpofed to be eafily reflected, or eafily tranf-
mittedj by every vibration which overtakes it. But

whether

{8ij

whether this Hypothefis be true or falic I do not here
tonfider. I content my felf with the bare difcovery,
that the rays of Light are by fome caufe or other alter-
nately difpofed to be refleded or refracted for many vi-
ciffitudes.

"D EFINITION.

The returns of the diffo/ition of any ray to be rejle^ed
I imll call its Fits of eafy reflexion, and thofe of
its dtffofition to be tranfmitted its Fits of eafy tranf-
miflion, and the fface it fajfes bet'ween every re-
turn and the next return^ the Interval of its
Fits.

PROP. XIII.

The reafon isuhy the fur faces of all thick tranf parent
Bodies refleH fart of the Light incident on them^ and
refra6l the refi^ is^ that fome rays at their incidence are
in Fits of eafy refiexion^ and others in Fits of eafy tranf-
mijjion.

This may be gathered from the a^th Obfervation,
where the Light reflected by thin plates of Air and Glafs,
which to the naked Eye appeared evenly white all over
the plate, did through a Prifm appear waved with many
fuccellions of Light and Darknefs made by alternate fits
of eafy reflexion and eafy tranfmiffion , the Prifm
fevering and diftinguifhing the waves of which the
white refled:ed Light was compofed, as was explained
above.

M m And

[82]

And hence Light is in fits of eafy reflexion and eafy
tranfmiffion, before its incidence on tranfparent Bodies.
And probably it is put into fuch fits at its firft emiffion
from luminous Bodies, and continues in them during
all its progrefs. For thefe fits are of a lafting Nature,
as will appear by the next part of this Book.

In this Propofition I fuppofe the tranfparent Bodies
to be thick, becaufe if the thicknefs of the Body be
much lefs than the interval of the fits of eafy reflexion
and tranfmiflion of the rays, the Body lofethits refleding
power. For if the rays, which at their entering into
the Body are put into fits of eafy tranfmiflion, arrive at
the furthefl; furface of the Body before they be out of
thofe fits they mufl: be tranfmitted. And this is the
reafon why Bubbles of Water lofe their reflecting power
when they grow very thin, and why all opake Bo'
dies when reduced into very fmall parts become tranf-
parent.

PROP. XIV.

T'hofe fur faces of tr an f -parent Bodies^ isuhich if the ra>j
he in a fit of refraction do refra^ it mofi firongly^ if the
my he in a Jit of reflexion do refleB it mofi eafily.

For we (hewed above in Prop. 8. that the caufe of
reflexion is not the impinging of Light on the folid
impervious parts of Bodies, but fome other power by
which thofe folid parts aft on Light at a diftance. We
fliewed alfo in Prop. 9. that Bodies refled: and refrad
Light by one and the fame power varioufly exercifed in
various circumfl:ances, and in Prop. i. that the moft
ftrongly refrading furfaces reflect the mofl: Light : All

which

[ 8? ]

which compared together evince and ratify both this
and the la ft Propofition.

PROP. XV.

In any one and the fame fort of rays emerging in a,ny
Jungle out of any refraBing fur face into one and the fame
medium^ the interval of the foUoisjing jits of eafy reflexion
and tranfmijfi-on are either accurately or very nearly^ as
the ReSlangle of the fecant of the Angle of refrailion^ and
of the fecant of another Angle ^ lajhofe fine ts the firfi of
1 06 arithmetical mean proportionals , letisjeen the fines
of incidence and refraB,ion counted from the fine of re-
fraB,ion.

This is manifeft by the 7th Obfervation.

PROP. XVI.

Jn fever al forts of rays emerging in equal Angles out
of any refraSling furface into the fam€ medium^ the inter*
vals of the foUoiioing jits of eafy reflexion and eafy tranf-
mijfion are either accurately^ or very nearly^ a^ the Cube*
roots of the Squares of the lengths of a Chord^ isjhich found
the notes in an Eighty fol, la, fa, fol, la, mi, fa, fol, with
all their intermediate decrees anfisjering to the Colours of
thofe rays^ according to the Analogy defended in the fe*
venth Experiment of the fecond Book.

This is manifeft by the 13 th and i^thObfervations.

Mm a PROR

[84]

PROP. XVII,

Ifra>js of an>j one fort fnfs -perfenclicularl'j into Jeveral
mediums^ the intervals of the jits of eafy reflexion and
tranfmijjlon in any one medium^ is to thofe intervals m
any other as the fine of incidence to the fine of refraHion^
isohen the rays fafs out of the firft of thofe two mediums
into the fecond.

This is manifeft by the loth Obfervation.

PROP. XVIIL

Jf the rays isuhich faint the Colom' tn the confine of
yelioisj and orange fafs fer-j^endicularly out of any medium
into Jlir^ the intervals of their fits of eafy refiexion are
the ^J:h fart of an Inch. And of the fame length are
the intervals of their fits of eafy tranfmijjlon.

This is manifeft by the 6th Obfervation.

From thefe Propofitions it is eafy to col]e(5l the in-
tervals of the fits of eafy reflexion and eafy tranfmif-
fion of any fort of rays refra(5ted in any Angle into
any medium, and thence to know, whether the rays
ftiall be refle(5ted or tranlmitted at their fubfequent
incidence upon any other pellucid medium. Which
thing being ufeflil for underftanding, the next part of
this Book was here to be fet down. And for the fame
teafoa I add the. two following Propofitions.

PROP.

[85]

PROP. XIX.

If any fort of rays falling on the polite fur face of any
pellucid medium he rejleHed hach^ the fits of eajy re^
jlexion isuhich they have at the point of reflexion , Jhall
jiill continue to return^ and the returns Jhall be at di-
jiances from the point of reflexion in the arithmetical
progrejjion of the numbers 2, 4, 6, 8, 10, la^&c. and be-
fween thefe fits the rays Jhall be in, fits of eafy tranf^
miffion.

For fince the fifS of eafy reflexion and eafy tranf-
miflion are of a returning nature, there is no reafon
why thefe fits, which continued till the ray arrived at
the reflecting medium, and there inclined the ray to
reflexion, fliould there ceafe. And if the ray at the
point of reflexion was in a fit of eafy reflexion, the
progreflion of the diftances of thefe fits from that point
muft begin from o, and lb be of the numbers o, 2, 4.,
6, 8, \^c. And therefore the progreflion of the di-
ftances of the intermediate fits of eafy tranfmiflion rec-
koned from the fame point, muft be in the progreflion
of the odd numbers i, 5, 5, 7, 9,155'^. contrary to what
happens when the fits are propagated from points o£
refraction.

PROP. XX.

The intervals of the fits of eafy reflexion and eafy
tranfmiffion^ propagated from points of reflexion into my
medium^ are eq^ual to the intervals of the like fits -which
the fame rays 'would have^ if refraBed into the fame

^** medium

medium in Angles of j'efraiHion equal to their j4ngles of
rejiexion ,

For when Light is refleded by the fecond furface of
thin plates, it goes out afterwards freely at the firft fur-
face to make the Rings of Colours which appear by
reflexion, and by the freedom of its egrefs, makes the
Colours of thefe Rings more vivid and ftrong than thofe
which appear on the other fide of the plates by the
tranfmitted Light. The reflected rays are therefore in
fits of eafy tranfmiflion at their egrefs ; which would
not always happen, if the intervals of the fits within
the plate after reflexion were not tqual both in length
and number to their intervals before it . And this confirms
alio the proportions fet down in the former Propofition.
For if the rays both in going in and out at the firft furface
be in fits of eafy tranfmiflion, and the intervals and num-
bers of thofe fits between the firft and fecond furface,
before and after reflexion, be equal ; the diftances or
the fits of eafy tranfmiflion from either furface, muft be
in the fame progreflion after reflexion as before ; that
is, from the firft furface which tranfmitted them, in
the progreflion of the even numbers o, a, 4, 6, 8, "^c.
and from the fecond which refle(fied them, in that of
the odd numbers i, 3, 5, 7, }S)c. But thefe two Pro-
pofitions will become much more evident by the Obfer*
vations in the following part of thisBooli.

THE

[87]
THE

SECOND BOOK

O F

O P T I C K S

PART IV.

Ohjervations concerning the Reflexions and Colours of
thick tranf^arent foltjhed 'Plates.

THere is no Glafs or Speculum how well foever
polifhed, but, belides the Light which it refrads
or reflects regularly , fcatters every way irregularly a
faint Light, by means of which the polifhed furface,
when illuminated in a dark Room by a beam of the
Sun's Light, may be eafily feen in all pofitions of the
Eye. There are certain Phgenomena of this fcattered
Light, which when I firft obferved them, feemed very
ftrange and furpriling to nie. My Obfervations were
as follows.

OBS.

[88]

O B S. I.

The Sun (hining into my darkened Chamber through
a Hole \ of an Inch wide, I let the intromitted beam
of Light fall perpendicularly upon a Glafs Speculum
ground concave on one fide and convex on the other,
to a Sphere of five Feet and eleven Inches Radius, and
quick'filvered over on the convex fide. And holding
a white opake Chart, or a Quire of Paper at the Center
of the Spheres to which the Speculum was ground, that
is, at the diftance of about five Feet and eleven Inches
from the Speculum, in fuch manner, that the beam of
Light might pafs through a little Hole made in the
middle of the Chart to the Speculum, and thence be
reflected back to the fame Hole : I obferved upon the
Chart four or five concentric Irifes or Rings of Colours,
like Rain-bows, encompaffing the Hole much after the
manner that thofe, which in the fourth and following
Obfervations of the firft part of this third Book appeared
between theObjed-Glaires,encompaffed the black Spot,
but yet larger and fainter than thofe. Thele Rings as
they grew larger and larger became diluter and fainter,
lb that the fifth was fcarce vifible. Yet fometimes,
when the Sun fhone very clear, there appeared faint
Lineaments of a fixth and feventh. If the diftance of
the Chart from the Speculum was much greater or much
lefs than that of fix Feet, the Rings became dilute and
vaniflied. And if the diftance of the Speculum from
the Window was much greater than that of fix Feet,
the refledted beam of Light would be fo broad at the
xliftance of fix Feet from the Speculum where the Rings

appeared,

C 8? 3

appeared, as to obfcure one or two of the innermoft
Rings. And therefore I ufually placed the Speculum
at about fix Feet from the Window ; fo that its Focus
might there fall in with the center of its concavity at the
Rings upon the Chart. And this pofture is always to
be underftood in the following Obl'ervations where no
other is expreft.

O B S. II.

The Colours of thefe Rain-bows fucceeded one ano^
ther from the center outwards, in the fame form and
order with thofe which were made in the ninth Obfer-
vation of the firft Part of this Book by Light not re-
fieded, but tranfmitted through the two Objeft-Glafles.
For, firft, there was in their common center a white
round Spot of faint Light, fomething broader than the
refleded beam of Light ; which beam fometimes fell
upon the middle of the Spot, and fometimes by a little
inclination of the Speculum receded from the middle,
and left the Spot white to the center.

This white Spot was immediately encompaffed with
a dark grey or ruffet, and that darknefs with the Co-
lours of the firft Iris, which were on the infide next
the darknefs a little violet and indico, and next to that
a blue, which on the outfide grew pale, and then fuc-
ceeded a little greenifh yellow, and after that a brighter
yellow, and then on the outward edge of the Iris a red
which on the outfide inclined to purple.

This Iris was immediately encompaffed with a fe-
cond, whofe Colours were in order from the infide

N n out-

[90]

outwards, purple, blue, green, yellow, light red, a red
mixed with purple.

Then immediately followed the Colours of the third
Iris, which were in order outwards a green inclining
to purple, a good green, and a red more bright than
that of the former Iris.

The fourth and fifth Iris feemed of a bluifh green
within, and red without, but fo faintly that it was dif-
ficult to difcern the Colours.

O B S. III.

Meafuring the Diameters of thefe Rings upon the
Chart as accurately as I could, I found them alfo in
the fame proportion to one another with the Rings
made by Light tranfmitted through the two Objed-
Glafles. For the Diameters of the four firft of the
bright Rings meafured between the brighteft parts of
their orbits, at the diftance of fix Feet from the Specu-
lum were ij^, a|, aji, 5I Inches, whofe fquares are in
arithmetical progreffion of the numbers i, a, ^, 4.. If
the white circular Spot in the middle be reckoned
amongft the Rings, and its central Light , where it
feems to be moft luminous, be put equipollent to an
infinitely little Ring ; the fquares of the Diameters of the
Rings will be in the progreflion o, i, a, 5, 4., l^c. I
meafured alfo the Diameters of the dark Circles be-
tween thefe luminous ones, and found their fquares
in the progrefiion of the numbers |, i', a-^, 3i, b'r.
the Diameters of the firft four at the diftance of fix Feet
from the Speculum, being i^^, 2;-^, i^^ ^f^ Inches. If
the diftance of the Chart from the Speculum was in-

crealed

creafed or dimlnifhed, the Diameters of the Circles were
iacrealed or diminifhed proportionally.

O B S. IV.

By the analogy between thefe Rings and thofe de-
fcribed in the Obfervations of the firftPart of this Book,
I fufpected that there were many more of them which
fpread into one another, and by interfering mixed their
Colours, and diluted one another fo that they could
not be feen apart. I viewed them therefore through a
Prifm, as I did thofe in the 2 4.th Obfervation of the
firft Part of this Book. And when the Prifm was fo
placed as by refracting the Light of their mixed Co*
lours to feparate them, and diftinguilh the Rings from
one another, as it did thofe in that Obfervation, I could
then fee them diftinder than before, and eafily num-
ber eight or nine of them, and fometimes twelve or
thirteen. And had not their Light been fo very faint,
I queftion not but that I might have feen many more.

O B S. V.

Placing a Prifm at the Window to refradl the intro-
mitted beam of Light, and caft the oblong Spectrum
of Colours on the Speculum : I covered the Speculum
with a black Paper which had in the middle of it a Hole
to let any one of the Colours pafs through to the Spe-
culum, whilft the reft were intercepted by the Paper.
And now I found Rings of that Colour only which fell
upon the Speculum. If the Speculum was illuminated
with red the Rings were totally red with dark inter-

Nn 2 vals,

[92]

vals, if with blue they were totally blue, and fo of the
other Colours. And when they were illuminated with
any one Colour, the Squares of their Diameters mea-
fured between their moft luminous parts, were in the
arithmetical progreffion of the numbers o, i , a, :5^ 4., and
the Squares of the Diameters of their dark intervals in
the progreffion of the intermediate numbers 7, i-l, i\, g^:
But if the Colour was varied they varied their magni-
tude. In the red they were largeft, in the indico and
violet leaft, and in the intermediate Colours yellow,
green and blue ; they were of feveral intermediate big-
neffes anfwering to the Colour, that is, greater in yel-
low than in green, and greater in green than in blue.
And hence I knew that when the Speculum was illumi-
nated with white Light, the red and yellow on the out-
fide of the Rings were produced by the leaft refrangible
rays, and the blue and violet by the moft refrangible,
and that the Colours of each Ring fpread into the Co-
lours of the neighbouring Rings on either fide, after
the manner explained in the firft and fecond Part of this
Book, and by mixing diluted one another fo that they
could not be diftinguiihedj unlefs near the center where
they were leaft mixed. For in this Obfervation I could
fee the Rings more diftindly, and to a greater number
than before, being able in the yellow Light to number
eight or nine of them, befides a faint ftiadow of a tenth.
To fatisfy my lelf how much the Colours of the feveral
Rings fpread into one another, I meafured the Diame-
ters of the fecond and third Rings , and found them
when made by the confine of the red and orange to be
the fame Diameters when made by the confine of blue
and indicoj as 9 to 8, or thereabouts. For it was hard

[93]

to determine this proportion accurately. Alfo the Cir-
cles made fucceffively by the red, yellow and green,
differed more from one another than thofe made fuccef-
fively by the green, blue and indico. For the Circle
made by the violet was too dark to be feen. To carry
on the computation, Let us therefore fuppofe that the
differences of the Diameters of the Circles made by the
outmoft red, the confine of red and orange, the confine
of orange and yellow, the confine of yellow and green,
the confine of green and blue, the confine of blue and
indico, the confine of indico and violet, and outmoft vio-
let, are in proportion as the differences of the lengths
of a Monochord which found the tones in an Eight ;
foljla^fa^fol^la^mi^fa^fol^ that is, as the numbers i,
1-8, IT, u, i7, '7, Is- -^^^<i i^ ^he Diameter of the Circle made
by the confine of red and orange be 9 A, and that of
the Circle made by the confine of blue and indico be
8 A as above, their difference 9 A — 8 A will be to
the difference of the Diameters of the Circles made by
the outmoft red, and by the confine of red and orange^,
as fs + T2 + U. + ir to 9, that is as T'^ to i or 8 to 5, and to
the difference of the Circles made by the outmoft violet,
and by the confine of blue and indico, as f s + f 2 + f » + 17
to i/ + ^s, that is, as I7 to h, or as 16 to 5. And there-
fore thefe differences will be f A and U A. Add the
firft to 9 A and fubdud the laft from 8 A, and you
will have the Diameters of the Circles made by the
leaft and moft refrangible rays s^ A and pi A. Thefe
Diameters are therefore to one another as 75 to 61^ or
50 to 41, and their Squares as 2500 to 1681, that is^
as ^ to 2 very nearly. Which proportion differs not
much from the proportion of the Diameters of the

Circles

[94]

Circles made by the outmoft red and outmoft violet in
the 1 3th Obfervation of the firft part of this Book.

O B S. VI.

Placing my Eye where thefe Rings appeared plaineft,
1 law the Speculum tinged all over with waves of Co-
lours ( red, yellow, green, blue ; ) like thole which in
the Obfervations of the firft Part of this Book appeared
between the Objed-Glaffes and upon Bubbles of Water,
but much larger. And after the manner of thofe, they
were of various magnitudes in various poiitions of the
Eye, fwelling and ftirinking as 1 moved my Eye this
way and that way. They were formed like Arcs of
concentrick Circles as thofe were, and when my Eye
was over againft the center of the concavity of the Spe-
culum (that is, 5 Feet and i o Inches diftance from the
Speculum) their common center was in a right Line
with that center of concavity, and with the Hole in the
Window. But in other poftures of my Eye their center
had other pofitions. They appeared by the Light of
the Clouds propagated to the Speculum through the
Hole in the Window, and when the Sun fhone through
that Hole upon the Speculum, his Light upon it was
of the Colour of the Ring whereon it fell, but by its
fplendor obfcured the Rings made by the Light of the
Clouds, unlefs when the Speculum was removed to a
great diftance from the Window, fo that his Light upon
it might be broad and faint. By varying the pofition of
my Eye, and moving it nearer to or farther from the
■dired beam of the Sun's Light, the Colour of the Sun's
reilesSed Light conftantly varied upon the Speculum,

[95 3

as it did upon my Eye, the fame Colour always ap-
pearing to a By-ftander upon my Eye which to me ap-
peared upon the Speculum. And thence I knew that
the Rings of Colours upon the Chart were made by thefe
reflected Colours propagated thither from the Specu-
lum in feveral Angles, and "that their production de-
pended not upon the termination of Light and Shad-
dow.

O B S. VIL

By the Analogy of all thefe Phasnomena with thofe of
the like Rings of Colours defcribed in the firft Part of
this Book, it feemed to me that thefe Colours were
produced by this thick plate of Glafs , much after the
manner that thofe were produced by very thin
plates. For, upon tryal, I found that if the Quick-
iilver were rubbed off from the back-fide of the Specu-
lum, the Glafs alone would caufe the fame Rings of
Colours, but much more faint than before ; and there-
fore the Phenomenon depends not upon the Quick-
filver, unlefs fo far as the Quick-filver by the increafing
the reflexion of the back-fide of the Glafs increafes the-
Light of the Rings of Colours. I found alfo tliat a Spe*
culum of metal without Glafs made fome years fince
for optical ufes, and very well wrought, produced none
of thofe Rings ; and thence I underftood that thefe
Rings arife not from one fpecular furface alone , but
depend upon the twofurfaces of the plate of Glafs where'
of the Speculum was made, and upon the thicknefs of
the Glafs between them. For as in the 7th and 1 9th
Obfervations of the firft Part of this Book a thin plate

-of Air, Water, or Glafs of an even thicknefs appeared
of one Colour when the rays were perpendicular to it,
of another when they were a little oblique, of another
when more oblique, of another when ftill more oblique,
and fo on ; fo here, in the^xth Obfervation, the Light
which emerged out of the Glals in feveral obliquities,
made the Glafs appear of feveral Colours, and being
propagated in thofe obliquities to the Chart, there pain-
ted Rings of thofe Colours. And as the reafon why a
thin plate appeared of feveral Colours in feveral obli-
quities of the rays,was,that the rays of one and the fame
Ibrt are refleded by the thin plate at one obliquity and
tranfmitted at another, and thofe of other forts tranf-
mitted where thefe are reflected, and reflected where
thefe are tranfmitted : So the reafon why the thick
plate of Glafs whereof the Speculum was made did ap-
pear of various Colours in various obliquities, and in
thofe obliquities propagated thofe Colours to the Chart,
was, that the rays of one and the fame fort did at one
obliquity emerge out of the Glafs, at another did not
emerge but were reflefted back towards the Quick-fil-
ver by the hither furface of the Glafs, and accordingly
as the obliquity became greater and greater emerged
and were retieded alternately for many fucceffions, and
that in one and the fame obliquity the rays of one fort
were refleded, and thofe of another tranfmitted. This
is manifeft by the firft Obfervat'.on of this Book : For
in that Obfervation, when the Speculum was illumi-
nated by any one of the prifmatick Colours, that Light
made many Rings of the fame Colour upon the Chart
w^ith dark intervals, and therefore at its emergence out
of the Speculum was alternately tranfmitted, and not

tranf-

[97]

tranfmltted from the Speculum to the Chart for many
fucceffions, according to the various obUquities of its
emergence. And when the Colour caft on the Specu-
lum by the Prifm was varied, the Rings became of
the Colour caft on it, and varied their bignefs with their
Colour, and therefore the Light was now alternately
tranfmitted and not tranfmitted from the Speculum to
the Lens at other obliquities than before. It feemed to
me therefore that thefe Rings were of one and the fame
original with thole of thin plates, but yet with this
difference that thofe of thin plates are made by the al-
ternate reflexions and tranfmiffions of the rays at the
fecond furface of the plate after one paffage through it :
But here the rays go twice through the plate before
they are alternately reflected and tranfmitted ; firft,
they go through it from the firft furface to the Quick-
filver, and then return through it from the Quick-filver
to the firft furface, and there are either tranfmitted to
the Chart or reflected back to the Quick-filver, ac-
cordingly as they are in their fits of eafie reflexion or
tranfmiffion when they arrive at that furface. For the
intervals of the fits of the rays which fall perpendicu-
larly on the Speculum, and are reflected back in the
fame perpendicular Lines, by reafon of the equality of
thefe Angles and Lines,are of the fame length and num-
ber within the Glafs after reflexion as before by the
19th Propofition of the third Part of this Book. And
therefore fince all the rays that enter through the firft
furface are in their fits of eafy tranfmiflion at their en-
trance, and as many of thefe as are reflected by the fe-
cond are in their fits of eafy reflexion there, all thefe
muft be again in their fits of eafy tranfmiflion at their

O o return

[98]

return to the firft, and by confcquence there go out of
the Glafs to the Chart, and form upon it the white
Spot of Light in the center of the Rings. For the rea-
fon holds good in all forts of rays , and therefore all
forts muft go out promifcuouily to that Spot, and by
their mixture cauie it to be white. But the intervals
of the fits of thofe rays which are refleded more ob-
liquely than they enter, muft be greater after reflexion
than before by the 15 th and aoth Prop. And thence
it may happen that the rays at their return to the firft
furface, may in certain obliquities be in fits of eafy re-
flexion, and return back to the Quick-filver, and in
other intermediate obliquities be again in fits of eafy
tranfmiffion, and fo go out to the Chart, and paint on
it the Rings of Colours about the white Spot. And
becaufe the intervals of the fits at equal obliquities are
greater and fewer in the lefs refrangible rays, and lefs
and more numerous in the more refrangible, therefore
the lels refrangible at equal obliquities fhall make fewer
Rings than the more refrangible, and the Rings, made
by thofe fhall be larger than the like number of Rings
madebythefe; that is, the red Rings fhall be larger
than the yellow, the yellow than the green, the green
than the blue, and the blue than the violet, as they
were really found to be in the 5th Obfervation. And
therefore the firft Ring of all Colours incompaffing the
white Spot of Light Ihall be red without and violet
within, and yellow, and green, and blue in the middle,
as it was found in the fecond Obfervation ; and thefe
Colours in the fecond Ring, and thofe that follow Ihall
be more expanded till they fpread into one another.
and blend one another by interfering.

Thefe

L99:i

Thefe feem to be the reafons of thefe Rings in ge-
neral, and this put me upon obferving the thicknefs of
the Glafs, and confidering whether the dimenfions and
proportions of the Rings may be truly derived from it
by computation.

O B S. VIII.

I meafured therefore the thicknefs of this concavo-
convex plate of Glafs, and found it every-w^here + of an
Inch precifely. Now, by the 6th Obfervation of the
firft Part of this Book, a thin plate of Air tranfmits the
brighteft Light of the firft Ring, that is the bright yel-
low, when its thicknefs is the 89000th part of an Inch,
and by the i oth Obfervation of the fame part, a thin
plate of Glafs tranfmits the fame Light of the fame Ring
when its thicknefs is lefs in proportion of the fine of
refraftion to the fine of incidence, that is, when its
thicknefs is the t^^^^ or .^^^th part of an Iiich, fup-
pofing the fines are as 11 to 17. And if this thicknefs
be doubled it tranfmits the fame bright Light of the'
fecond Ring, if tripled it tranfmits that of the third',
and fo on, the bright yellow Light in all thefe cafes be-
ing in its fits of tranfmiffion. And therefore if its thick-
nefs be multiplied 34-386 times fo as to become \ of an
Inch it tranfmits the fame bright Light of the 34386th
Ring. Suppofe this be the bright yellow Light tranf-
mitted perpendicularly from the reileding convex fide
tDf the Glafs through the concave fide to the white Spot
in the center of the Rings of Colours on the Chart : And
by a rule in the feventh Obfervation in the firft Part of
the firft Book, and by the 1 5 th and a oth Propofitions

O o ci of

[lOO]

of the third Part of this Book, if the rays be made ob-
lique to the Glals, the thicknefs of the Glafs requi-
lite to tranfmit the fame bright Light of the fame Ring
in any obliquity is to this thicknefs of } of an Inch, as
the fecant of an Angle whofe line is the firll: of an hun-
dred and fix arithmetical means between the fines of
incidence and refraction, counted from the fine of inci-
dence when the refradion is niade out of any plated Bo-
dy into any medium incompaffing it, that is, in this cafe,
out of Glafs into Air. Now if the thicknefs of the Glafs
be increafed by degrees,fo as to bear to its firft thicknefs,
( viz. that of a quarter of an Inch ) the proportions
which 34386 (the number of fits of the perpendicular
rays in going through the Glafs towards the white Spot
in the center of the Rings,) hath to 34.385, 34-384,
34383 and 3438a (the numbers of thefits of the oblique
rays in going through the Glafs towards the firft, fe-
cond, third and fourth Rings of Colours,) and if the
firft thicknefs be divided into 1 00000000 equal parts,
the increafed thicknefles will be 100002908, 100005816,
100008725 and 100011635^ and the Angles of which thefe
thicknefles are fecants will be 26' 13", 37' 5", 45' 6" and
52' 16", the Radius being 1 00000000 ; and the fines of
thefe Angles are 76a, 1079, 1311 and 1525, and the
proportional fines ofrefradion 1172, 1659, 2031 and
2345, the Radius being 1 00000. For fince the fines
of incidence out of Glafs into Air are to the fines
of refraftion as 11 to 1 7, and to the above-mentioned
fecants as 1 1 to the firft of 106 arithmetical means
between 11 and 17, that is as 11 to ii.fe, thofe fe-
cants will be to the fines of refraction as iif^fito 17,
and by this Analogy will give thefe fines. So then

if the obliquities of the rays to the concave llirfdce of
the Glafs be fuch that the fines of their refradtion in
paffing out of |the Glafs through that furface into the
Air be 1172, 1659, ^ogi, 2^4-5, the bright Light of
the 54.^ 86th Ring fhall emerge at the thickneffes of the
Glafs which are to \ of an Inch as 34-^86 to 34-385,
34384., 34.383, 34.382, refpedively. And therefore if
the thicknefs in all thefe cafes be^ of an Inch (as it is in
the Glafs of which the Speculum was made) the bright
Light of the 34.385th Ring fliall emerge where the line
of refraction is 1 17a, and that of the 3^4384.th, 384.383th
and 34.381th Ring where the fine is 1659, 2031, and
234.5 refpeCtively. And in thefe Angles of refradioii
the Light of thefe Rings fhall be propagated from the
Speculum to the Chart, and there paint Rings about the
white central round Spot of Light which we laid was
the Light of the 34.386th Ring. And the Semidiame-
ters of thefe Rings fhall fubtend the Angles of refradion,
made at the concave furface of the Speculum, and by
confequence their Diameters fhall be to the diftance of
the Chart from the Speculum as thole fines of refradion-
doubled, are to the Radius that is as 1 171, 1659, 2031,
and 234.5, doubled are to 1 00000. And therefore if
the diilance of the Chart from the concave furface of
the Speculum be fix Feet (as it was in the third of thefe
Obfervations) the Diameters of the Rings of this bright
yellow Light upon the Chart fhall be I'^SS, 2*3 89,,
2*925) 3'375 Inchti. : For thefe Diameters are to 6 Feet
as the above-mentioned fines doubled are to the Radius.
Now thefe Diameters of the bright yellow Rings, thus
found by computation are the very fame with thofe
found in the third of thefe Obfervations by meafuring

them.

C 102 ]

them^ (vtx. with i|i> a^* i'7,and ^'-Inches, and there-
fore the Theory of deriving thefe Rings from the thick-
nefa of the plate of Glafs of which the Specuktm was
made, and from the obliquity of the emerging rays agrees
with the Obfervation. In this computation I have
equalled the Diameters ot the bright Rings made by
Light of all Colours, to the Diameters of the Rings
made by the bright yellow. For this yellow makes the
brightell part of the Rings of all Colours. If you defire
the Diameters of the Rings made by the Light of any
other unmixed Colour, you may find them readily by
putting them to the Diameters ot the bright yellow ones
in a fubduplicate proportion of the intervals of the fits
of the rays of thole Colours when equally inclined to
the refrading or reflecting furface which caufed thofe
fits, that is, by putting the Diameters of the Rings made
by the rays in the extremities and limits of the feven
Colours, red, orange, yellow, green, blue, indico, violet,
proportional the Cube-roots of the numbers, i , f , 6 ' J ,
Mo ?6J "U which exprefs the lengths of a Monochard'
founding the notes in an Eight : For by this means the
Diameter of the Rings of thefe Colours will be found
pretty nearly in the fame proportion to one another,
which they ought to have by the fifth of thefe Obfer-
vations.

And thus I fatisfied my felf that thefe Rings were of
the fame kind and original with thofe of thin plates,
and by confequence that the fits or alternate difpofi-
tions of the rays to be refieded and tranfmitted are pro-
pagated to great diftances from every refleding and re-
frading furface. But yet to put the matter out of doubt
I added the following Obfervation.

OBS.

[ I03 ]

O B S. IX.

If thefeRIogs thus depend on the thicknefs of the plate
of Glafs their i)i3meters at equal diftances from feveral
Speculums made of fuch concavO'Convex plates of Glafs
as are ground on the fame Sphere, ought to be recipro-
cally in a fubduplicate proportion of the thicknefTes of
the plates of Glafs. And if this proportion be found
true by experience it will amount to a demonftration
that thefe Rings ( like thofe formed in thin plates ) do
depend on the thicknefs of the Glafs. I procured there-
fore another concavo-convex plate of Glafs ground on
both fides to the fame Sphere with the former plate ::
Its thicknefs was |, parts of an Inch ; and the Diameters
of the three firft bright Rings meafured between the
brighteft parts of their orbits at the diftance of 6 Feet
from the Glafs were 5. 4^. 5^. Inches. Now the thick-
nefs of the other Glafs being \ of an Inch was to thicks
nefs of this Glafs as \ to i, , that is as ^ i to i o, or
310000000 to 1 00000000^ and the roots of thefe numbers
are 17607 and loooo, & in the proportion of the firft
of thefe roots to the fecond are the Diameters of the^
bright Rings made in this Obfervation by the thinner.
Glafs, 3. 4^. 5^ to the Diameters of the fame Rings mader
in the third of thefe Obfervations by the thicker Glafs
i{]. a-' aj^, that is, the Diameters of the Rings arerecir-
procally in a fubduplicate proportion of thicknefTes of
the plates of Glafs.

So then in plates of Glafs which are alike concave on
one fide, and alike convex on the other fide, and alike
quick-filvered on the convex fides, and ditfer in nothing,^

but.

[104]

but their thicknefs, the Diameters of the Rings are re-
ciprocally in a fubduplicate proportion of the thicknefles
of the plates. And this fhews lufficiently that the Rings
depend on both the furfaces of the Glafs. They de-
pend on the convex furface becaufe they are more lu-
minous when that furface is quick-filvered over than
when it is without Quick-filver. They depend alfo
upon the concave furface, becaufe without that furface
a Speculum makes them not. They depend on both
furfaces and on the diftances between them , becaufe
their bignefs is varied by varying only that diftance.
And this dependance is of the fame kind with that
which the Colours of thin plates have on the diftance
of the furfaces of thofe plates , becaufe the bignefs
of the Rings and their proportion to one another,
and the variation of their bignefs arifing from the varia-
tion of the thicknefs of the Glafs, and the orders of
their Colours, is fuch as ought to refult from the Propo*
litions in the end of the third Part of this Book, derived
from the the Phaenomena of the Colours of thin plates
fet down in the firft Part.

There are yet other Phaenomena of thefe Rings of
Colours but fuch as follow from the fame Propofitions,
and therefore confirm both the truth of thofe Propofi-
tions, and the Analogy between thefe Rings and the
Rings of Colours made by very thin plates. I fhall
fubjoyn fome of them.

O B S.

O B S. X.

When the beam of the Sun's Light was refleded back
from the Speculum not diredtly to the Hole in the Win-
dow, but to a place a little diftant from it, the common
center of that Spot, and of all the Rings of Colours fell
in the middle way between the beam of the incident
Light, and the beam of the refleded Light, and by
confequence in the center of the fpherical concavity of
the Speculum, whenever the Chart on which the Rings
of Colours fell was placed at that center. And as the
beam of refleded Light by inclining the Speculum re-
ceded more and more from the beam of incident Light
and from the common center of the coloured Rings be-
tween them, thofe Rings grew bigger and bigger, and
lb alfo did the white round Spot, and new Rings of Co-
lours emerged fucceflively out of their common center,
a*id the white Spot became a white Ring encompafling
them ; and the incident and refleded beams of Light
always fell upon the oppoflte parts of this Ring, illumi-
nating its perimeter like two mock Suns in the oppoflte
parts of an iris. So then the Diameter of this Ring,
meafured from the middle of its Light on one fide to
the middle of its Light on the other fide, was always
equal to the diftance between the middle of the incident
beam of Light, and the middle of the refleded beam
meafured at the Chart on which the Rings appeared :
And the rays which formed this Ring were refleded by
the Speculum in Angles equal to their Angles of inci-
dence, and by conlequence to their Angles of refradion
at their entrance into the Glafs, but yet their Angles of

P p reflexion

reflexion were not in the fame planes with their Angles
of incidence.

O B S. XI.

The Colours of the new Rings were in a contrary
order to thofe of the former, and arofe after this man-
ner. The white round Spot of Light in the middle of
the Rings continued white to the center till the diftance
of the incident ond reflected beams at the chart was
about I parts of an Inch, and then it began to grow
dark in the middle. And when that diftance was about
1^6 of an Inch, the white Spot was become a Ring en-
compaffing a dark round Spot which in the middle in-
clined to violet and indico. And the luminous Rings
incompaffing it were grown equal to thofe dark ones
which in the four firft Obfervations encompafled them,
that is to fay, the white Spot was grown a white Ring
equal to the firft of thofe dark Rings, and the firft t)f
thofe luminous Rings was now grown equal to the fe-
cond of thofe dark ones, and the fecond of thofe lumi-
nous ones to the third of thofe dark ones, and fo on.
For the Diameters of the luminous Rings were now 1,7,,

^re, ^p Vto,'^''- Inches.

When the diftance between the incident and refleded
beams of Light became a little bigger, there emerged
out of the middle of the dark Spot after the indico a
blue, and then out of that blue a pale green, and foon
after a yellow and red. And when the Colour at the
center was brighteft, being between yellow and red,
the bright Rings were grown equal to thofe Rings which
in the tour firft Obfervations next encompafled them ;

that

[107]

that is to fay, the white Spot in the middle of thofe
Rings was now become a white Ring equal to the firft
of thofe bright Rings, and the firft of thofe bright ones
was now become equal to the fecond of thofe, andfo
on. For the Diameters of the white Rings, and of the
other luminous Rings incompaffing it, were now lil,
28, 2i'i J ^8, J5'<:-. or thereabouts.

When thediftance of the two beams of Light at the
Chart was a little more increafed, there emerged out
of the middle in order after the red, a purple, a blue,
a green, a yellow, and a red inclining much to purple,
and when the Colour was brighteft being between yel-
low and red, the former indico, blue, green, yellow and
red, were become an Iris or Ring of Colours equal
to the firft of thofe luminous Rings which appeared in
the four firft Obfervations, and the white Ring which
was now become the fecond of the luminous Rings was
grown equal to the fecond of thofe, and the firft of
thofe which was now become the third Ring was be-
come the third of thofe, and fo on. For their Diame-
ters were 1^6, ai, arf, ^f Inches, the diftance of the
two beams of Light, and the Diameter of the white
Ring being a^ Inches.

When thefe two beams became more diftant there
emerged out of the middle of the purplifti red, firft a
darker round Spot, and then out ot the middle of that
Spot a brighter. And now the former Colours (purple,
blue, green, yellow, and purpliili red ) were become a
Ring equal to the firft of the bright Rings mentioned in
the four firft Obfervations , and the Rmg about this
Ring were grown equal to the Rings about that re^
fpedively ; the diftance between the two beams of

P p 2 Light

[io8]

Cight and the Diameter of the white Ring ( which
was now become the third Ring ) being about ^ In-
ches.

The Colours of the Rings in the middle began now
to grow very dilute, and if the diftance between the
two beams was increafed half an Inch, or an Inch more,
they vanifhed whilft the white Ring, with one or two
of the Rings next it on either tide, continued ftill vi-
able. But if the diftance of the two beams of Light
was ftill more increafed thefe alfo vanifhed : For the
Light which coming from feveral parts of the Hole in
the Window fell upon the Speculum in feveral Angles of
• incidence made Rings of feveral bignefles, which diluted
and blotted out one another, as I knew by intercepting
fome part of that Light. For if 1 intercepted that part
which was neareft to the Axis of the Speculum the
Rings would be lefs, if the other part which was re-
moteft from it they would be bigger.

O B S. XIL

When the Colours of the Prifm were caft fucceffively
on the Speculum, that Ring which in the two laft Ob-
fervations was white, was of the fame bignefs in all the
Colours, but the Rings without it were greater in the
green than in the blue, and ftill greater in the yellow^
and greateft in the red. And, on the contrary, the
Rings within that white Circle were lefs in the green
than in the blue, and ftill lefs in the yellow, and leaft
in the red. For the Angles of reflexion of thofe rays
which made this Ring being equal to their Angles of
incidence, the fits of every retieded ray within the Glafs

after

[lOp]

after reflexion are equal in length and number to the
fits of the fame ray within the Glafs before its incidence
on the reflecting furface ; and therefore fince all the rays
of all forts at their entrance into the Glafs were in a fit
of tranfmiflion, they were alfo in a fit of tranfmiffion at
their returning to the fame furface after reflexion ; and
by confequence were tranfmitted and went out to the
white Ring on the Chart. This is the reafon why that
Ring was of the fame bignefs in all the Colours, and
why in a mixture of all it appears white. But in rays
which are refleded in other Angles, the intervals of the
fits of the leafl: refrangible being greatefl:, make the
Rings of their Colour in their progrels from this white
Ring, either outwards or inwards, increafe or decreafe
by the greatefl: fteps ; fo that the Rings of this Colour
without are greatefl, and within leaft. And this is the
reafon why in the laft Obfervation, when the Specu-
lum was iUuminated with white Light, the exterior
Rings made by all Colours appeared red without and
blue within, and the interior blue without and red
within.

Thefe are the Phsenomena of thick convexo-concave
plates of Glafs, which are every where of the fame
thicknefs. There are yet other Phsenomena when thefe
plates are a little thicker on one fide than on the
other, and others when the plates are more or lefs con-
cave than convex, or plano-convex, or double-convex.
For in all thefe cafes the plates make Rings of Colours,
but after various manners j all which, fo far as I have
yet obferved, follow from the Propofitions in the end
of the third part of this Book, and fo confpire to con^
firm the. truth of thofe Propofitions. But the Phseno-

mena.

[no]

mena are too various, and the Calculations whereby
they follow from thole Propofitions too intricate to be
here profecuted. I content my felf with having profe-
cuted this kind of Phaenomena lb far as to difcover their
caufe, and by difcovering it to ratify the Propoiitions
in the third Part of this Book.

O B S. XIII.

As Light reliecled by a Lens quick-lilvered on the
back'lide makes the Rings of Colours above de-
icribed, fo it ought to make the like Rings of Colours
in palTing through a drop of Water. At the lirft re-
flexion of the rays within the drop, fome Colours ought
to be tranfmitted, as in the cafe of a Lens, and others
to be reflected back to the Eye. For inftance, if the
Diameter of a fmall drop or globule of Water be about
the 500th part of an Inch, fo that a red-making ray in
pafling through the middle of this globule has 250 fits
of eafy tranfmiffion within the globule, and that all the
red-making rays which are at a certain diftance from
this middle ray round about it have 14.9 fits within the
globule, and all the like rays at a certain further di-
liance round about it have 148 fits, and all thofe at a
certain further difiiance ^24.7 fits, and fo on ; thefe con-
centrick Circles of rays after their tranfmiffion, falling
on a white Paper, will make concentrick rings of red
upon the Paper , fuppofing the Light which pafles
through one lingle globule Itrong enough to be fenfible.
And, in like manner, the rays of other Colours will
make Rings of other Colours. Suppofe now that in a
fair day the Sun Ihines through a thin Cloud of fuch

globules

[HI]

globules of Water or Hail, and that the globules are all
of the fame bignefs,and the Sun feen through this Cloud
(hall appear incoinpaffed with the like concentrick Rings
of Colours, and the Diameter of the firll Ring of red
fhall be 7; degrees, that of the fecond i O; degrees, that
of the third 12 degrees ^^ minutes. And accordingly
as the globules of Water are bigger or lefs, the Rings
fhall be lefs or bigger. This is the Theory, and expe-
rience anfwers it. For in yune 1691. 1 law by reflexion
in a Veflel of ftagnating Water tliree Halos Crowns or
Rings of Colours about the Sun, like three little Rain-
bows, concentrick to his Body. The Colours of the
firft or innermoft Crown were blue next the Sun, red
without, and white in the middle between the blue
and red. Thofe of the fecond Crown were purple and
blue within, and pale red without, and green in the
middle. And thole of the third were pale blue with-
in, and pale red without ; thele Crov/ns inclofed one
another immediately, fo that their Colours proceeded
in this continual order from the Sun outward : blue,
white, red ; purple, blue, green, pale yellow and red ;
pale blue, pale red. The Diameter of the fecond Crown
meafured from the middle of the yellow and red on one
fide of the Sun, to the middle of the fame Colour on
the other fide was 9^ degrees, or thereabouts. The Dia*
meters of the firft and third 1 had not time to meafure,
but that of the firft leemed to be about five or fix de-
grees, and that of the third about twelve. The like
Crowns appear fometimes about the Moon ; for in the
beginning of the year 1 664, Fek\ 1 9th at night, I faw
two fuch Crowns about her. The Diameter of the firft
or innermoft was about three degrees, and that of the.

fecond

[112]

fecond about five degrees and an half. Next about the
Moon was a Circle of white, iand next about that the
inner Crown which was of a bluiili green within next the
white, and of a yellow an4 red without, and next about
thefe Colours were blue and green on the infide of the
outward Crown, and red on the outlide of it. At the
lame time there appeared a Halo about 11 degrees 35'
diftant from the center of the Moon. It was Elli]-)rical,
and its long Diameter was perpendicular to the Kuiizon
verging below fartheft from the Moon. I am told that
the Moon has fometimes three or more concentrick
Crowns of Colours incompafling one another next about
her Body. The more equal the globules of Water or
Ice are to one another, the more Crowns of Colours
will appear, and the Colours will be the more lively.
The Halo at the diftance of 11^ degrees from the Moon
is of another fort. By its being oval and remoter from
the Moon below than above, 1 conclude, that it was
made by refradion in fome fort of Hail or Snow floating
in the Air in an horizontal Pofture, the refra6\inc!, Angle
bemg about 58 or 60 degrees.

THE

BoOK,n. Plate.I.

"%,.

%.%

hikl

clnncp^jn ti'xyz.

^ ^ SfS- g g" s

Fig 1.

„. BooK,lI.'. Plate,!.

Fig: 2.

^j^'

Zy X v t s r ^ pcnmlLih gfed c

dcf^kik.lntai'p^rs ti'xyz.

OL 13 y d

B C B E F G H

i4i M M

Book.n Plate.n.

Ltr.b.

,/.'■ .■■// .4/ , X / / •

Fig^ 7.

["3]

THE

THIRD BOOK

O F

O P T I C K S.

Objervations concerning the Inflexions of the rays of Light ^
and the Colours made thereby.

GKimaldo has informed us, that if a beam of the
Sun's Light be let into a dark Room through a
very fmall Hole, the Ihadows of things in this Light
will be larger than they ought to be if the rays went
on by the Bodies in ftreight Lines, and that thefe fha-
dows have three parallel fringes, bands or ranks of cO'
loured Light adjacent to them. But if the Hole be
enlarged the fringes grow broad and run into one ano-
ther, lb that they cannot be diftinguilhed. Thefe broad
Ihadows and fringes have been reckoned by fome to pro-
ceed from, the ordinary refradlion of the Air, but with-
out due examination of the matter. For the circum-
ftances of the Phaenomenon, lb far as I have obferved
them, are as follows.

Q q O B S,

O B S. I.

1 made in a piece of Lead a Imall Hole with a Pin,
whofe breadth was the4^th part of an Inch. For ii
of thofePins laid together took up the breadth of half
an Inch. Through this Hole 1 let into my darkened
Chamber a beam of the Sun's Light, and found that the
fliadowsofHairs,Thred,Pins,Straws, and fuchlike {len-
der fubftances placed in this beam of Light, were conlider-
abiy broader than they ought to be, if the rays of Light
paffed on by thefe Bodies in right Lines. And particu-
larly a Hair of a Man's. Head, whole breadth was but
the 280th part of an Inch, being held in this Light, at
the diftance of about twelve Feet from the Hole, did
caft a fliadow which at the diftance of four Inches from
the Hair was the fixtieth part of an Inch broad, that is,
above four times broader than the Hair, and at the di-
ftance of tw^o Feet from the Hair was about the eight
and twentieth part of an Inch broad, that is, ten times
broader than the Hair,^ and at the diftance often Feet
w^as the eighth part of an Inch broad, that is 5 5 times
broader.

Nor is it material whether the Hair be incompaffed
with Air, or with any other pellucid fubltance. For I
wetted a poliflied plate of Glafs, and laid the Hair in
the Water upon the Glafs, and then laying another po-
lifhed plate of Glafs upon it, lb that the Water might
lill up the fpace between the GlalTes, I held them in
the aforeGid beam of Light, fo that the Light might
pals through them perpendicularly, and the lliadow
of the Hair was at the lame ditfances as big as before.

The

C"5]

The (hadows of fcratches made in poliflied plates of
Glafs were alfo much broader than they ought to be,
and the Veins in poHflied plates of Glafs did alfo caft the
like broad fhadows. And therefore the great breadth
of thefe fhadows proceeds from fome other caufe than
the refraftion of the Air.

Let the Circle X reprefent the middle of the Hair; Fig. i.
ADG, BEH, CFI, three rays pa ffing by one fide of
the Hair at feveral diftances ; KNQ, LOR, MPS,
three other rays paffing by the other fide of the Hair at
the like diftances; D, E, F and N, O, P, the places
where the rays are bent in their paffage by the Hair ;
G, H, I and Q, R, S, the places where the rays fall on
a Paper G Q ; 1 S the breadth of the fiiadow of the Hair
caft on the Paper, and T I, V S, two rays paffing to the
points I and S without bending when the Hair is taken
away. And it's manifeft that all the Light between
thefe two rays A I and V S is bent in paffing by the
Hair, and turned afide from the fliadow IS, becaule if
any part of this Light were not bent it would fall on
the Paper within the fhadow, and there illuminate the
Paper contrary to experience. And becaufe when the
Paper is at a great diftance from the Hair, the fhadow
is broad, and therefore the rays T I and V S arc at a
great diftance from one another, it follows that the
Hairafts upon the rays of Light at a good diftance in
their paffing by it. But the action is ftrongeft on the
rays which pafs by at leaft diftances, and grows weaker
and weaker accordingly as the rays pafs by at diftances
greater and greater, as is reprcfented in the Scheme :
For thence it comes to pais, that the Ihadow of the
Hair is much broader in proportion to the diftance of

Qq 2 the

[11(5]

the Paper from the Hair, when the Paper is nearer the
Hair than when it is at a great diftance from it.

O B S. 11.

The fhadows of all Bodies ( Metals, Stones, Glafs,
Wood, Horn, Ice, Id'c\ ) in this Light were bordered
with three parallel fringes or bands of coloured Light,
whereof that which was contiguous to the Ihadow was
broadeft and moft luminous, and that which was re-
moteft from it was narroweft, and fo faint, as not eafily
to be vifible. It was difficult to diiHnguilh the Colours
unlefs when the Light fell very obliquely upon a fmooth
Paper, or fome other fmooth white Body, lb as to make
them appear much broader than they would otherwife
do. And then the Colours were plainly vliible in this
order : The firft or innermoft fringe was violet and deep
blue next the fhadow, and then light blue, green and
yellow in the middle, and red without. The fecond
fringe was almoft contiguous to the firft, and the third
to the fecond, and both were blue within and yellow
and red without, but their Colours were very faint
efpecially thofe of the third. The Colours therefore
proceeded in this order from the fhadow, violet, indico,
pale blue, green, yellow, red ; blue, yellow, red ; pale
blue, pale yellow and red. The ihadows made by
fcratches and bubbles in polilhed plates of Glafs were
bordered with the like fringes of coloured Light. And
if plates of Looking'glafs floop'd off near the edges with
a Diamond cut, be held in the fame beam of Light, the
Light which paffes through the parallel planes of the
Glafs will be be bordered with the like fringes of Co-
lours

["7]

lours where thole Planes meet with the Diamond cut,
and by this means there will ibmetimes appear four or
five fringes of Colours. Let AB, CD reprefent th^Fig. 2.
parallel planes of a Looking-glafs, and BD the plane
of the Diamond-cut, making at B a very obtuie Angle
with the plane A B. And let all the Light between the
rays EN I and FBM pafs diredly through the parallel
planes of the Glafs, and fall upon the Paper between I
and M, and all the Light between the rays G O and
HD be refracted by the oblique plane of the Diamond
cut B D,and fall upon the Paper between K and L ; and
the Light which pafles diredly through the parallel
planes of the Glafs, and falls upon the Paper between
I and M, will be bordered with three or more fringes
at M.

o B s. in.

When the Hair was twelve Feet diftant from the-
Hole, and its fhadow fell obliquely upon a flat white
fcale of Inches and parts of an Inch placed half a Foot
beyond it, and alfo when the fliadow fell perpendicu-
larly upon the fame fcale placed nine Feet beyond it;
I meafured the breadth of the fhadow and fringes as
accurately as I could, and found them in parts of ari;
Inch as follows.

The

["8]

At the dijlance of

half A

Foot.

The breadth of the Shadow

The breadth between the middles of the
brighteft Light of the innermoft fringes
on either iide the fhadow

The breadth between the middles of the
brighteft Light of the middlemoft frin-
ges on either lide the fhadow

The breadth between the middles of the
brighteft Light of the outmoft fringes
on either lide the ftiadow

The diftance between the middles of the
brighteft Light of the firft and lecond
fringes

The diftance between the middles of the
brighteft Light of the lecond and third
fringes

;? 01- ;;

23^

75 or }..,

1S2

120

The breadth of the luminous part (green,
white, yellow and red ) of the firft
fringe

The breadth of the darker Ipace between
the firft and lecond fringes.

The breadth of the luminous part of the
lecond fringe

The breadth of the darker fpace between
the fecond and third fringes.

J 70

nine

Feet.

7.
50

4_
»7

r
31

Thefe

[119]

Thefe meafures I took by letting the fliadovv of the
Hair at half a Foot diftance fall fo obliquely on the
fcale as to appear twelve times broader than when it
fell perpendicularly on it at the fame diftance, and fet-
ting down in this Table the twelfth part of the mea-
fures I then took.

O B S. IV.

When the fhadovv and fringes were caft obliquely
upon a fmooth white Body, and that Body was remo-
ved further and further from the Hair, the lirft fringe
began to appear and look brighter than the reft of the
Light at the diftance of lefs than a quarter of an Inch
from the Hair, and the dark line or ftiadovv between
that and the fecond fringe began to appear at a lefs di-
ftance from the Hair than that of the third part of an
Inch. The fecond fringe began to appear at a diftance
from the Hair of lefs than half an Inch, and the ftiadow
between that and the third fringe at a diftance lefs than
an Inch, and the third fringe at a diftance lefs than three
Inches. At greater diftances they became much jnore
ienfible, but kept very nearly the fame proportion of
their breadths and intervals which they had at their ftrft
appearing. For the diftance between the middle of the
firft and middle of the fecond fringe, was to the diftance
between the middle of the fecond and middle of the
third fringe, as three to two, or ten to feven. And
the laft of thefe two diftances was equal to the breadtli
of the bright Light or luminous part of the hrft fringe^
And this breadth was to the breadth of the bright Light
of the fecond fringe as feven to tour, and to the dark

interval!

[112]

interval of the firft and lecond fringe as three to two,
and to the like dark interval between the fecond and
third as two to one. For the breadths of the fringes
feemed to be in the progreffion of the numbers i, f^'-^
l^) and their intervals to be in the fame progreffion
'with tliem ; that is, the fringes and their intervals to-
gether to be in the continual progreffion of the numbers
I, /{, f^\, l^\, //-', or thereabouts. And thefe pro-
portions held the fame very nearly at all dirtances from
the Hair ; the dark Intervals of the fringes being as
broad in proportion to the fringes at their firft appea-
rance as afterwards at great diftances from the Hair,
thoui?h not fo dark and diftind.

O B S. V.

The Sun fhining into my darkened Chamber through
a Hole a quarter of an Inch broad ; I placed at the di-
ftance of two or three Feet from the Hole a Sheet of
Paft-board, which was black'd all over on both fides,
and in the middle of it had a Hole about three quarters
of an Inch fquare for the Light to pafs through. And
behind the Hole I fattened tothePatl-board with Pitch
the blade of a Iharp Knife, to intercept fome part of
the Light which paffed through the Hole. The planes
of the Paft'board and blade of the Knife were parallel
to one another, and perpendicular to the rays. And
when they were fo placed that none of the Sun's Light
fell on the Paft-board, but all of it paffed through the
Hole to the Knife, and there part of it fell upon the
blade of the Knife, and part of it paffed by its edge :
I let this part of the Light which paffed by, fall on a

white

C.I2I]

white Paper two or three Feet beyond the Knife, and
there faw two ftreams of faint Light ihoot out both
ways from the beam of Light into thefliadow hke the
tails of Comets. But becaufe the Sun's diredl Light by
its brightnefs upon the Paper obfcured thefe faint
ftreams, fo that 1 could fcarce fee them, I made a little
Hole in the midft of the Paper for that Light to pafs
through and fall on a black cloth behind it ; and then
I faw the two ftreams plainly. They were like one
another, and pretty nearly equal in length and breadth,
and quantity of Light. Their Light at that end next
the Sun's diredl: Light was pretty ftrong for the fpace of
about a quarter of an Inch, or half an Inch, and in all
its progrefs from that dired Light decreafed gradually
till it became infenhble. The whole length of either of
thefe ftreams meafured upon the Paper at the diftance
of three Feet from the Knife was about fix or eight
Inches ; fo that it fubtended an Angle at the edge of
the Knife of about lo or 12, or at moft 14. degrees.
Yet fometimes I thought I faw it ftioot three or four
degrees further, but with a Light fo very faint that I
could fcarce perceive it, and fufpeded it might ( in
fomjC meafure at leaft) arife from fome other caufe than
the two ftreams did. For placing my Eye in that Light
beyond the end of tliat ftream which was behind the
Knife, and looking towards the Knife, I could fee a
line of Light upon its edge, and that not only when
my Eye was in the line of the ftreams, but alio when
it was without that line either towards the point of the
Knife, or towards the handle. This line of Light ap-
peared contiguous to the edge of the Knife, and was
narrower than the Light of the iimermoft fringe, and

R r narroweft

[122]

narrowcft when my Eye was furtfieft from the dlreft
Light, and therefore Teemed to pafs between the Light
of that fringe and the edge of the Knife, and that
whicli p.ifTed ncareft the edge to be moft bent, though
not all of it.

O B S. Vl.

I placed another Knife by this fo that their edges
might be parallel and look towards one another, and
that the beam of Light might fall upon both the Knives,
and Ibme part of it pals between their edges. And
when the diftance of their edges was about the 4.00th
part of an Inch the ftream parted in the middle, and
left a iTiadow between the two parts. This fhadow
was fo black and dark that all the Light which paffed
between the Knives feemed to be bent, and turned afide
to the one hand or to the other. And as the Knives ftill
approached one another the fhadow grew broader, and
the ftreams fhorter at their inward ends which were
next the iliadow, until upon the contadt of the Knives
the whole Light vanifhed leaving its place to the
fhadow.

And hence I gather that the Light which is leaft
bent, and goes to the inward ends of the ftreams, paf-
fes by the edges of the Knives at the greateft diftance,
and this diftance when the fhadow begins to appear be-
tween the ftreams is about the eight- hundredth part of
an Inch. And the Light which paffes by the edges of
the Knives at diftances ftill lefs and lets is more and
more bent, and goes to thofe parts of the ftreams which
are further and further from the direft Light, becaufc

when

[123]

when the Knives approach one another till they touch,
thole parts of the ftreams vaniih laft which are furtheft
from the dired Liglit.

O B S. VII.

In the fifth Obfervation the fringes did not appear,
but by reafon of the breadth of the Hole in the Win-
dow became fo broad as to run into one another, and
by joyning make one continued Light in the beginning
of the ftreams. But in the iixth, as the Knives ap-
proached one another, a little before the fhadow ap-
peared between the two ftreams, the fringes began to
appear on the inner ends of the ftreams on either fide
of the dire6l Light, three on one fide made by the edge
of one Knife, and three on the other fide made by the
edge of the other Knife. They were diftindeft when
the Knives were placed at the greateft diftance from the
Hole in the Window, and ftill became more diftind by
making the Hole lefs, infomuch that I could fometimes
fee a faint lineament of a fourth fringe beyond the three
above-mentioned. And as the Kniv^es continually ap-
proached one another, the fringes grew diftinder and
larger until they vaniftied. The outmoft fringe va-
niftied firft, and the middlemoft next, and the inner-
moft laft. And after they were all vaniftied, and the
line of Light which was in the middle between them
was grown very broad, enlarging it felf on both fides
into the ftreams of Light defcribed in the fifth Obfer-
vation, the above-mentioned Ihadow began to appear
in the middle of this line, and divide it along the middle
into two lines of Light, and increafcd until the whole

Rr a Light

Light vaniflied. This inlargement of the fringes was
fo great that the rays which go to the innermoft fringe
feemed to be bent above twenty time's more when this
fringe was ready to vanifh, than when one of the Knives
was taken away.

And from this and the former Obfervation compared,
I gather, that the Light of the firft fringe paffed by the
edge of the Knife at a diftance greater than the eight-
hundredth part of an Inch,, and the Light of the fecond
fringe pafled by the edge of the Knife at a greater di-
ftance than the Light of the firft fringe did, and that
of the third at a greater diftance than that of the fe-
cond, and that of the ftreams of Light defcribed in
the fifth and fixth Obfervations palled by the edges
of the Knives at Icfs diftances than that of any of the
fringes.

O B S. VIII.

I caufed the edges of two Knives to be ground truly
ftreight, and pricking their points into a board fo that
their edges might look towards one another, and meet-
ing near their points contain a re6lilinear Angle, I faft-
ned their handles together with Pitch to make this
Angle invariable. The diftance of the edges of the
Kpives from one another at the diftance of four Inches
from the angular point, where the edges of the Knives
met, was the eighth part of an Inch, and therefore the
Angle contained by the edges was about i degr. 54'.
The Knives thus fixed together I placed in a beam of
the Sun's Light, let into my darkened Chamber througli
a Hole the ^ith part of an Inch wide, at the diftance

[125]

of ten or fifteen Feet from th^ Hole, and let the Light
which paffed between their edges fall very obliquely
upon a fmooth white Ruler at the diftance of half an
Inch, or an Inch from the Knives, and there faw the
fringes made by the two edges of the Knives run along
the edges of the (hadows of the Knives in lines parallel
to thole edges without growing leniibly broader, till
they met in Angles equal to the Angle contained by the
edges of the ^nives, and where they met and joyned
they ended without croffing one another. But if the
Ruler was held at a much greater diftance from the
Paper,the fringes became fomething broader and broader
as they approached one another, and after they met
they crofled one another, and then became much broader
than before.

Whence I gather that the diftances at which the
fringes pais by the Knives are not increafed nor altered
by the approach of the Knives, but the Angles in which
the rays are there bent are much increafed by that ap-
proach J and that the Knife which is neareft any ray
determines which way the ray lliall be bent, and the
other Knife increafes the bent.

O B S. IX.

When the rays fell very obliquely upon the Ruler at
the diftance of the third part of an Inch from the Knives,
the dark line between the firft and lecond fringe of the
fhadow of one Knife, and the dark line between the
firft and fecond fringe of the fhadow of the other Knife
met with one another, at the diftance of the fifth part
of an Inch from the end of the Light which pafled be-
tween,

[126]

tween the Knives at the concourfe of their edges. And
therefore the diftance of the edges of the Knives at the
meeting of thefe dark lines was the i6oth part of an
Inch. For as four Inches to the eighth part of an Inch,
fo is any length of the edges of the Knives rneaibred
from the point of their concourfe to the diftance of the
edges of the Knives at the end of that length, and fo is
the hfth part of an Inch to the 1 6oth part. So then the
dark lines abo^^e-mcntioned meet in the middle of the
Light which paffes between the Knives where they are
diftant the i6othpartof an Inch, and the one half of
that Light pafles by the edge of one Knife at a diftance
not greater than the 520th part of an Inch, and falling
upon the Paper makes the fringes of thelliadow of that
Knife, and the other half palTes by the edge of the
other Knife, at a diftance not greater than the ^lotli
part of an Indi, and falling upon the Paper makes the
ifringes of the ftiadow of the other Knife. But if the
Paper be held at a diftance from the Knives greater than
the third part of an Inch, the dark lines above-men-
tioned meet at a greater diftance than the fifth part of
an Inch from the end of the Light which palled be-
tween the Knives at the concourfe of their edges; and
therefore the Light which falls upon the Paper where
thofe dark lines meet pafles between the Knives
where their edges are diftant above the i6oth part of
an 'Imdh.

For at another time when the two Knives were di-
ftant eight Feet and five Indies from the little Hole in
the Window, made with a finall Pin as above, the Light
which fell upon the Paper where the aforefaid dark
lines met. pafled between tlie Knives, where the di-
ftance

[127]

ftance between their edges was as in the followincv
Table, when the diftance of the Paper from the Knive°
was alio as follows.

Dijiances bf the Paper
from the Kjitves in
Inches.

3^'

96.

131.

Dijiances between the edges
of the Kjiives in mille-
[tm at farts of an Inch.

o oia.

0'0 20.

o'o54..
©'057.
o'o8i.

And hence I gather that the Light which makes the
fringes upon the Paper is not the lame Light at all di-
ftances of the Paper from the Knives, but when the Pa-
per is held near the Knives, the fringes are made bv
Light which paffes by the edges of the Knives at a lels
diftance, and is more bent than when the Paper is held
at a greater diftance from tlie Knives.

O B S. X.

When the fringes of the ftiadows of the Knives fell
perpendicularly upon a Paper at a great diftance from
the Knives, they were in the form of Hyperbolas, and
their dimenfions were ^s follows. LetCA, CBrepre-
fent lines drawn upon the Paper parallel to the edges of
the Knives, and between which all the Light w^ould
fall, if it palled between the edges of the Knives with-
out inflexion; DE a right line drawn through C making

the

[128] ■

the Angles A CD, BCE, equal to one another, and
terminating all tlie Light whith falls upon the Paper from
the point where the edges of the Knives meet ; eis, fk t,
and g 1 V, three hyperbolical lines reprefenting the ter-
minus ofthe iliadowofoneof the Knives, the dark line
between the firll and fecond fringes of that fhadow, and
the dark line between the fecond and third fringes of
the fame fhadow ; x i p, y k q and z 1 r, three other Hy-
perbolical lines reprefenting the terminus of the fhadow
of the other Knife, the dark line between the firft and
fecond fringes of that fhadow, and the dark line be-
tween the fecond and third fringes of the fame ihadow.
And conceive that thefe three Hyperbolas are like and
equal to the former three, and crofs them in the points
i, k and 1, and that the fhadows of the Knives are termi-
nated and diftinguifhed from the firft luminous fringes
by the lines e is and xip, until the meeting and crof-
ling of the fringes, and then thofe lines crofs the fringes
in the form of dark lines, terminating the firft luminous
fringes within fide, and diftinguifhing them from ano-
ther Light wliich begins to appear at i, and illuminates
all the triangular fpace ipDEs comprehended by thefe
clark lines, and the right line DE. Of thefe Hy-
perbolas one Afymptote is the line DE, and their other
Afymptotes are parallel to the lines CA and CB. Let
rv reprefent a line drawn any where upon the Paper
parallel to the Afymptote D E, and let this line crofs
the right lines A C in m and B C in n, and the fix dark
hyperbolical lines in p, q, r ; s, t, v ; and by meafuring
the diftances ps, qt, r v, and thence collefting the
the, lengths of the ordinatesnp, nq, nr or ms, mt,
m V, and doing this at feveral diftances of the line r v,

from

[129]

from the Afymptote DE you may find as many points
of thefe Hyperbolas as you pleafe, and thereby know
that thefe curve lines are Hyperbolas differing little from
the conical Hyperbola. And by meafuring the lines
C i , C k , CI, you may find other points of thefe
Curves.

For inftance, when the Knives were diftant from the
Hole in the Window ten Feet, and the Paper from the
Knives 9 Feet, and the Angle contained by the edges of
the Knives to which the Angle ACB is equal, was fub'
tended by a chord which was to the Radius as i to ^a,
and the diftance of the line rv from the Afymptote DE
was half an Inch: I meafured the lines ps, qt, rv,
and found them o'^ 5, o'65, o'^S Inches refpedively,
and by adding to their halfs the line { mn (which here
was the 128th part of an Inch, or o'ooyS Inches ) the
fums np, nq, nr, were o'lSiS, o'^p8, o'^(^'j^ In-
ches. 1 meafured alfo the diftances of the brighteft
parts of the fringes which run between pq and st, qr
and t V, and next beyond r and v, and found them ©'5,
o'8, and Tiy Inches.

O B S. XI.

The Sun fhining into my darkened Room through a
fmall round Hole made in a plate of Lead with a llender
Pin as above ; I placed at the Hole a Prifm to refradl
the Light, and form on the oppofite Wall the Spectrum
of Colours, defcribed in the third Experiment of the
firft Book. And then I found that the fhadows of all
Bodies held in the coloured Light between the Prifm
and the Wall, were bordered with fringes of the Colour

S s of

[130]

of that Light in whicli they were held. In the full red

Light they were totally red without any fenfible blue

or violet, and in the deep blue Light they were totally

blue without any fenfible red or yellow ; and ib in the

green Light they were totally green, excepting a little

yellow and blue, which were mixed in the green Light

ofthePrifm. And comparing the fringes made in the

leveral coloured Lights^ 1 found that thofe made in the

red Light were largeft, thole made in the violet were

leaft, and thofe made in the green were of a middle

bianefs. For the fringes with which the fhadow of a

Man's Hair were bordered, being meafured crofs the

fhadow at the diftance of fix Inches from the Hair j the

diftance between the middle and moft luminous part of

the firft or innermoft fringe on one lide of the fliadow,

and that of the like fringe on the other lide of the Iha-

dow, was in the full red Light ^/- of an Inch, and in

the full violet ^. And the like diftance between the

middle and moft luminous parts of the fecond fringes on

either lide the fliadow was in the full red Light i, , and

in the violet '- of an Inch. And thefe diftances of the

fringes held the fiime proportion at all diftances from

the Hair without any fenlible variation.

So then the rays which made thefe fringes in the red
Light pafl'ed by the Hair at a greater diftance than thofe
did which made the like fringes in the violet; and there-
fore the Hair in caufing thefe fringes adf ed alike upon
the red Light or leaft refrangible rays at a greater di-
ftance, and upon the violet or moft refrangible rays at
a lefs diftance, and by thofe anions difpoied the red
Light into larger fringes, and the violet into fmaller,
and the Lights of intermediate Colours into fringes of

niter-

intermediate bignefles without changing thf Colour o^
of any Ibrt of Light.

When therefore the Hair in the firft and fecond of
thefe Obfervations was held in the white beam of the
Sun's Light, and caft a fhadow which was bordered with
three fringes of coloured Light, thofe Colours arofe not
from any new modifications impreft upon the rays of
Light by the Hair, but only from the various inflections
whereby the feveral forts of rays were feparated from
one another, which before feparation by the mixture
of all their Colours, compofed the white beam of the
Sun's Light, but w^ienever fepajated compofe Lights
of the feveral Colours which they are originally dil'po-
fed to exhibit. In this i ^th Obfervation, where the
Colours are feparated before the Light paffes by the
Hair, the lealt refrangible rays, which when fepara-
ted from the reft make red, were inflefted at a greater
diftance from the Hair, lb as to make three red fringes
at a greater diftance from the middle of the ftiadow of
the Hair 3 and the moft refrangible rays which when
feparated make violet, w^re intiefted at a lefs diftance
from the Hair, fo as to make three violet fringes at a
lefs diftance from the middle of the fhadow of the Hair.
And other rays of intermediate degrees of refrangibi-
lity were iniieded at intermediate diftances from the
Hair, fo as to make fringes of intermediate Colours at
intermediate diftances from the middle of the fhadow
of the Hair. And in the fecond Obfervation, where
all the Colours are mixed in the white Light which
paffes by the Hair, thefe Colours are feparated by the
various inflexions of the rays, and the fringes which
they make appear ail together, and the innermoft

Ss :: fringes

[132]

fringes being contiguous make one broad fringe compo-
fed of all the Colours in due order, the violet lying
on the infide of the fringe next the fhadow, the red on
the outfide furtheft from the fhadow, and the blue,
green and yellow, in the middle. And, in like man-
ner, the middlemoft fringes of all the Colours lying in
order, and being contiguous, make another broad fringe
compofed of all the Colours ; and the outmoft fringes
of all the Colours lying in order, and being contiguous,
make a third broad fringe compofed of all the Colours,
Thefe are the three fringes of coloured LIglit with'
which the (hadows of all Bodies are bordered in the le-
cond Obfervation.

When I made the foregoing Obfervations, I defigned
to repeat moft of them with more care and exadnefs,
and to make fome new ones for determining the man-
ner how the rays of Light are bent in their paflage by
Bodies for making the fringes of Colours with the
dark lines between them. But I was then interrup-
ted, and cannot now think of taking thele things inta
further conlideration.* And lince 1 have not Hnifhed
this part of my Defign, I fhall conclude, with propo-
fing only fome Queries in order to a further fearch to
be made by others,

^ery i . Do not Bodies ad upon Light at a diftance,.
and by their action bend its rays, and is not this atlion
(ceteris fariim) ftrongeft at the leaft dift:ance ?

^. 1. Do not the rays which differ in refrangibility
differ alfo in flexibility, and are they not by their dif-
ferent inflexions feparated from one another , fo as
after feparation to make the Colours in the three fringes

above

above defcribed ? And after what manner are they in-
flected to make thofe fringes ?

^. 5 . Are not the rays of Light in pafling by the
edges and (ides of Bodies, bent feveral times backwards
and forwards, with a motion Hke that of an Eel ? And
do not the three fringes of coloured Light above-men-
tioned, arife from three fuch bendings ?

^. 4. Do not the rays of Light which fall upon Bo-
dies, and are reflected or refradted, begin to bend be-
fore they arrive at the Bodies ; and are they not re-
fle(fled, refracted and infleded by one and the lame
Principle, acting varioufly in various circumdances?

.^. 5. Do not Bodies and Light a6l mutually upon
one another, that is to fay. Bodies upon Light in emit-'
ting, reflefting, refrading and infleding it, and Light
upon Bodies for heating them, and putting their parts
into a vibrating motion wherein heat conlifts ?

<^«. 6. Do not black Bodies conceive heat more eafily
from Light than thofe of other Colours do, by reafon
that the Light falling on them is not rcfleded outwards^
but enters the Bodies, and is often refledled and re-^ -
fradted within them, until it be ftifled and loft ?

^«. 7. Is not the ftrength and vigor of the a6lion
between Light and fulphureous Bodies obferved above^
one reafon why fulphureous Bodies take fire more
readily, and burn more vehemently , then other Bo-,
dies do ?

^. 8. Do not all fixt Bodies when heated beyond a
certain degree, emit Light and Ihine, and is not this
emiffion performed by the xibrating motions of their
parts ?

C134]

ti. 9. Is not fire a Body heated ib hot as to emit
Light copiouOy ? For what elfe is a red hot Iron than
fire ? And what elfe is a burning Coal than red hot
Wood?

^. 10. Is not flame a vapour, fume or exhalation
heated red hot, that is, fo hot as tolhine? For Bodies
do not flame without emitting a copious fume, and this ,
fume burns in the flame. The Jgnii Fatum is a vapour
fliining without heat, and is there not the fame differ
rence between this vapour and flame, as between rot-
ten Wood fliining without heat and burning Coals of
fire? In difl:ining hot Spirits, if the head of the fl:ill be
taken otf, the vapour which afcends out of the Still will
take fire at the flame of a Candle, and turn into flame,
and the flame wifl run along the vapour from the Candle
to the Still. Some Bodies heated by motion or fermen-
tation, if the heat grow intenfe fume copioufly, and if
the heat be great enough the fumes will fliine and be-
come flame. Metals in fufion do not flame for want of
a copious fume, except Spelter which fumes copioufly,
and thereby flames. All flaming Bodies, as Oyl, Tal-
low, Wax, Wood, foflil Coals, Pitch, Sulphur, by
flaming wafte and vanifli into burning fmoke,- which
fmoke, if the flame be put out, is very thick and vifible,
and fometimes fmells ftrongly, but in the flame lofcs
its fmell by burning, and according to the nature of the
fmoke the flame is of feveral Colours, as that of Sul-
phur blue, that of Copper opened with Sublimate
green, that of Tallow yellow. Smoke pafling through
flame cannot but grow red hot, and red hot imoke can
Jiave no other appearance than that of flame.

^u. I I .

[135]

^i. 1 1 . Do not great Bodies conferve their heat the
longeft, their parts heating one another, and may not
great denfe and fix'd Bodies, when heated beyond a
certain degree, emit Light fo copioufly, as by the e'mif-
fion and reaction of its Light, and the reflexions and re-
fractions of its rays within it? pores to grow ftill hot-
ter, till It COiTicS to a certain period of heat, fuch as is
that of the Sun ? And are not the Sun and fix'd Stars
great Earths vehemently hot, whofe heat is conferred
by the greatnefs of the Bodies, and the mutual action
and reatf ion between them, and the Light which they
emit, and whofe parts are kept from fuming away, not
only by their fixity, but alfo by the vaft weight and
denfity of the Atmofpheres incumbent upon them, and
very ftrongly compreffing them, and condenfing the va^
pours and exhalations which arife from them ?

^4. II. Do not the rays of Light in falling upon the
bottom of the Eye excite vibrations in the 'Vmica re^
tina ? Which vibrations, being propagated along the
folid fibres of the optick Nerves into the Brain, caufe
the fenfe of feeing. For becaufe denfe Bodies conferve
their heat a long time, and the denfeft Bodies conferve
their heat the longeft, the vibrations of their parts are
of a lafting nature, and therefore may be propagated
along folid fibres of uniform denfe matter to a great di-
fiance, for conveying into the Brain the impreffions
made upon all the Organs of fenfe. For that motion
which can continue long in one and rhe fame part of a
Body, can be propagated a long way from one part to
another, fujpoiing the Body homogeneal, fo that the
motion may not be retleded, refraded, interrupted or
difordered by any unevennefs of the Body.

«$«. I g . Do not feveral fort of rays make vibrations
of feveral bigneflcs, which according to their bignefl'es
excite fenlations of feveral Colours, much after the
manner that the vibrations of the Air, according to their
feveral bignefTes excite fenfations of feveral founds ?
And particularly do not the moft refrangible rays ex-
cite the fhorteft vibrations for making a fenfation of
deep violet, the leaft refrangible the largeft for making
a fenfation of deep red, and the feveral intermediate
Ibrts of rays, vibrations of feveral intermediate bignef-
fes to make fenfations of the feveral intermediate Co-
lours ?

^. 1 4. May not the harmony and difcord of Co-
lours ariie from the proportions of the vibrations propa-
gated through the fibres of the optick Nerves into the
Brain, as the harmony and difcord of founds arifes from
the proportions of the vibrations of the Air ? For fome
Colours are agreeable, as thofe of Gold and Indico, and
others difagree.

^. 15. Are not the Species of Objedts feen with both
Eyes united where the optick Nerves meet before
they come into the Brain, the fibres on the right fide
of both Nerves uniting there, and after union going
thence into the Brain in the Nerve which is on the
right fide of the Head, and the fibres on the left fide
of both Nerves uniting in the fame place, and after
union going into the Brain in the Nerve which is on
the left fide of the Head, and thefe two Nerves meet-
ing in the Brain in fuch a manner that their fibres
make but one entire Species or Pidure, half of which
on the right fide of the Senforium comes from the
right fide of both Eyes through the right fide of

both

[137]

both optick Nerves to the place where the Nerves
meet, and from thence on the right fide of the Head
into the Brain, and the other half on the left fide of the
Senforium comes in like manner from the left fide of
both Eyes. For the optick Nerves of fuch Animals as
look the lame way with both Eyes ( as of Men, Dogs,
Sheep, Oxen, }Sc. ) meet before they come into the
Brain, but the optick Nerves of fuch Animals as do
not look the fame way with both Eyes (as of Filhes and
of the Chameleon) do not meet, if I am rightly in-
formed.

^i. 1 6. When a Man in the dark prefles either cor-
ner of his Eye with his Finger, and turns his Eye away
from his Finger, he will fee a Circle of Colours like
thofe in the Feather of a Peacock's Tail ? Do not thefe
Colours arife from fuch motions excited in the bottom
of the Eye by the preffure of the Finger, as at other
times are excited there by Light for cauling Vifion ? And
when a Man by a ftroke upon his Eye fees a Flalh of
Light, are not the like Motions excited in the Retina
by the ftroke i^

[i38>

ENUMERATIO

LINEARUM

TERTII ORDINIS.

['39 3

ENUME RATIO

LINEARUM

TERTII ORDINIS.

LIneae Geometrical fecundum numerum diinen- i.
fionum squationis qua relatio inter Ordinatas ^i^j"'^^"* ^^'
Si Abfcififas definitur, vel (quod perinde eft) fecuri'
dum numerum pundorum in quibus a linea refta
fecari pofTunt, optime diftinguuntur in Ordines.
Qua ratione linea primi Ordinis erit Reda fola, ex
fecundi five quadratici ordinis erunt fediones Conicae
& Circulus, & ex tertii five cubici Ordinis Parabola
Cubica, Parabola Neiliana, Ciffois veterum & reli-
quas quas hie enumerare fuicepimus. Curva autem
primi generis, (fiquidem reda inter Curvas non eft
numeranda) eadem eft cum Linea fecundi Ordinis,
&. Curva fecundi generis eadem cum Linea Ordinis
tertii. Et Linea Ordinis infinitefimi ea eft quam
reda in pundtis infinitis fecare poteft, qualis eft Spi-
ralis, Cyclois, Quadratrix & linea omnis quae per
radii vel rotae revolutiones infinitas generatur.

Tt 2 Sedionum

[14°]

11- Seftionum Conicarum proprietates praeclpuif a?

^/oS' c"«7c/-' Geomctris paffini traduntur. Etconfiiniles lunt pro-
rum competuNt prietatesCurvaruiTi fecundi generis & reliquarum, ut
'^ffe7erfm""'^"'^ ^^ lequeiiti proprictatuiTi prascipuanrin cnumera-
tione conftabit.
in. Nam fi re6t?e plures psrallel<E & ad conicam le-

Curvarum fe- (f^jonem iitriiiq; termlnata? ducantur, reda duas ca-
d7LujDJlme-' ram bifecans bilecabit alias omnes,ideoq; dicitur 'DzVi--
^ri,f^ertkes^Cef!' y^gfgf {{gui'g^ ^ rcftoe bifettse dicLintur Oniinatim ap-
tra^Axes. ^Ucat.t: ad Diametrum, & concurfus omnium Dia-
metrorum eilCenU'tmi figurse, & interle6tio Curvae &
diametri Vertex nominatur, & diameter ilia Axis
eft Gui ordinatim applicatae infiftunt ad angulos re-
ctos. Et ad eundem modum in Curvis fecundi ge-
neris,, fi redae duas quaevis parallelae ducantur occur-
rentes Curvae in tribus pundis i, recta quae ita fecat
has parallelas ut lumma duar.um partium ex uno fe-
cantis latere ad curvam termiiiatarum aequctur parti
tertiae ex altero latere ad curvam terminata?, eodem
modo fecablt omnes alias his parallelas curvaeq; ia
tribus punttis occurrentes redas, hoc eft, ita ut lum-
ma partium duarum ex uno ipfius latere femper
aequetur parti tertiae ex altero latere. Has itaq- tres.
partes quae hinc inde a^quantur, Ordinatim a-ppli"
iTfttrti & reftam fecantem cui ordinatim applicantur
diametrum. 8l interfedionem diametri & curvae /^er-
ticem Sc concurfum duarum diametrorum Centrum.
nominare licet. Diameter autem ad Ordinatas re-
dangula fi modo aliqua fit, etiam Axis dici poteft,^
&. ubi omnes diametri in eodem pundo concurrunt

iliud erit. Centrum generate..

Hyper-

Hyperbola primi generis duas ^fympoi-os^ ea fe- iv:.
cundi tres,ea tertii quatuor & non plures habere. ^o^ea^'^'J^yi,!^^
tell, & iic in reliquis. Et quemadmodum partes, fe-^.
iinecE cujufvis redse inter Hyperbolam Conicam &
duas ejus Alymptotos lunt hinc indeaqualcs : fic in
Hyperbolis lecundi generis fi ducatur reda qucevis
fecans tarn Curvam quam tres ejus Afyinptotos in.
tribus pundis, lumma duarum partium iltius redse
quse a duobus quibulvis Alymptotis in eandem pla-
gara ad duo punda Curvs extcnduntur sequalis erir
parti tertiae quas a tertia Afymptoto in plagam con-
trariam ad tertlum Curva? pundium extenditur.

Et quemadmodum inConicis fedionibus non Pa- v.
rabolicis quadratum Ordinatim applicatae^ hoc efl, Laterargm.&
redangulum Ordinatarum quae ad contrarias par- ^''"^^^''^'''^
tes Diametri ducuntur, eft ad redangulum partium ■
Diametri quse ad Vertices Ellipieos vel Hyperbolae
terminantur,ut data qua:dam linea quce dicitur luatm
reHum^ ad partem diametri quae inter Vertices jacet
& dicitur Lattts tranfverfum : lie in Curvis non Para-
bolicis lecundi generis Parallelepipedum fub tribus-
Ordinatim applicatis eft ad Parallelepipedum fub par-
tibus Diametri ad Ordinatas & tres Vertices figures ab-
fciffis,,in ratione quadam data : in qua ratione fi lu-
mantur tres redae ad tres partes diametri inter ver-
tices figure fitas fingulas ad fingulas, tunc illse tres-
redix dici poffunt Latera reMo. iigurae, & ills? partes-
Diametri inter Vertices Latera tranfverja,. Et ficut
in Parabola Conica quce ad unam & eandem diame-
trum unicum tantum habet Verticem, redangulum
fubOrdinatis aequatur redangulolub parte Diametri
qux ad. Ordinatas & Verticem abfcLnditur & reda

q,uadam

quadam data quae Latus redum dicitur,fic in Curvis
lecundi generis qu3S non nifi duos habent Vertices ad
eandemDiametrum, ParallelepipedumfubOrdinatis
tribus aequatur Parallelepipedo Tub duabus partibus
Diametri ad Ordinatas & Vertices illosduos abfciffis,
& reda quadam data quGS proinde Latm redum
dici poteft.
VI. Deniq; ficut in Conicis fectionibus ubi duae paral-

y.^fulTaralie- ^^^^ ad Curvani utrinq; terminata? fecantur a dua-
lanimfegmemis. bus parallclis ad Curvam utrinq; terminatis, prima
a tertia & lecunda a quarta, re^tangulum partium
primce eft ad reftangulum partium tertiae ut redtan-
gulum partium fecunds ad reftangulum partium
quartse: iic ubi quatuor tales redos occurruntCurvae
fecundi generis fingulae in tribus punftis, parallele-
pipedum partium primae redae ei'it ad parallelepide^
. dum partium tertiae, ut parallelepipedum partium
fecundae ad parallelepipedum partium quarta?.
VII- Curvarum fecundi & fuperiorum generum aeque

^^/JJ'J2^P^^2^Iatq; primi crura omnia in infinitum progredientia
liM&eorumfia- vel Hfperholtci fuut geueris vel TaraMia. Crus Hy^
^'* ferholicum voco quod ad Afymptoton aliquam in in-

finitum appropinquat, Pttrai^(9/2<:7<?w quod Afymptoto
deftituitur. Hacc crura ex tangentibus optime dig'
nolcuntur. Nam fi pundlum contadus in infinitum
abeat tangens cruris Hyperbolici cum Afymptoto
coincidet Sc tangens cruris Parabolici in infinitum
recedet, evanefcet & nuUibi reperietur. Invenitur
igitur Afymptotos cruris cujufvis quaerendo tangen-
tem cruris illius ad pundum infinite diftans. Plaga
autem cruris infiniti invenitur quaerendo pofitionem
redae cujufvis quae tangenti parallela eit ubi pun-

dum

ChsB

ftum conta6tus in infinitum abit. Nam hcec redta
in eandem plagam cum crure inlinito dirigitur.

Linear omnes Ordinis primi, tertii, quinti, fep- viii.
timi & imparis cujufq; duo habent ad minimum Jlt^^Z^"^^
crura in infinitum verfus plagas oppofitas pvogYe^ generis fecmdud
dientia. Et lines omnes tertii Ordinis duo habent ''^"^''''""'Vf^
ejuimodi crura m plagas oppolitas progredicntia m primus.
quas nulla alia earum crura infinita (prseterquam
in Parabola Cartehana ) tendunt. Si crura ilia
fint Hyperbolici generis , fit G A S eorum Afymp-

totos ik. huic parallela agatur redta qua^vis CBc .^

ad Curvam utrinque ( fi fieri poteft ) terminata
eademq; biiecetur in punftoX, & locus pundi iX-H- ^'
lius X erit Hyperbola Conica ( puta X * ) cujus
una Afymptotos eft A S. Sit ejus altera Afymp-
totos A B, & cequatio qua relatio inter Ordinatam
BC & AbfcilTam AB definitur, fi AB dicatur x &
B C y, Temper induet banc formam xyy-|-ey = ax'
-(-bxx-^cx-l-d. Ubi termini e, a, b, c, d, defig-
nant quantitates datas cum fignis liiis -j-Sc— ' affe-
6las,quarum quaelibet deeflTe pofluntmodo ex earum
defedu figura in fedionem conicam non vertatur^ .
Poteft autem Hyberbola ilia Conica cum afympto-
tis fuis coincidere , id eft pundum X in re(5ta A B
locari : & tunc terminus -|-ey deeft.

At fi reda ilia CBc non poteft utrinq; ad Curvam
terminari led Curvae in unico tantum pun6lo occur- r r ^f' j
rit : age quamvis pofitione datam redam A B afymp- ^^^J^^"
toto AS occurrentem in A, ut & aliam quamvis BC
afymptoto illi parallelam Curvseque occurrentem in
pUii^toC, & aequatio qua relatio inter Ordinatam

BC:

B C Sc AblchTam A B detinitur, femper Induct hanc
formam x y = a x' -1^ b x x -|- c x -j- d.

X. Quod (i crura ilia oppolita Parabolici lint generis,
jjustertiM. j^jCj.^ CBcad Curvaui utrinque, li fieri potelt, ter-

minata in plagam crurum ducatur & bilecetur in B,
& locus pundTi B erit linea reda. Sit ifta AB, ter-
minata ad datum quodvis pun(5tum A, & aequatio
qua relatio inter Ordinatam BC & AbrciiTam AB
detinitur, Temper induct hanc formam, yy = ax^
-l-bxx4-cx-|-d.

XI. At vero ii redla ilia CB c in unico tantum pundo
'C^Hs quarttis. occurrat Curvas, idcoq; ad Curvam utrinq; terminari

non poffit : lit pun6tum illud C, & incidat re£ta ilia
ad punftum B in redam quamvis aliam poiitione
datam & ad datum quodvis pundum A terminatam
A B : (Sc squatio qua relatio inter Ordinatam B C &
Abfciiram AC detinitur femper induct hanc formam,
y = a x^-l~b X x-j- c x-|- d.
xii. Enumerando curvas horum cafuum, Hyperbolam

^Nommajorma' vocabimus infcfiptam quae tota jacet in Afymptoton
angulo ad inftar Hyperbolae conicae, circumfcnpam
quae Afymptotos fecat & partes abfciiTas in linu fuo
ampleditur, . ambigenam quae uno crure intinito in-
fcribitur & altero circumfcribitur , convergenxem
cujus crura concavitate fua feinvicem refpiciunt &
in plagam eandem ^^inguntuv ^dive^-gent em cujus crura
convexitate fua feinvicem recipiunt &. in plagas con-
trarias diriguntur, cruribus contrariis p^ccditam cujus
crura in partes contrarias convexa funt & in plagas
contrarias intinita, Conchoidalem quae vertice concave
& cruribus divcrgentibus ad afymptoton applicatur,^
( anguineam quae flexibus contrariis afymptoton fecat

CH5]

& utrinq; In crura contraria producitur, cmciformem
qua: conjugatam decuflat, nodatam quae felpfam de-
cuffat in orbem redeuiido, cuffidatam cujus partes
dus in angulo contadus concurrunt & ibi terminan-
tur, funSIatani quae conjugatam habet Ovalem infi-
nite parvam id eft pundum, & furam quae per im-
poflibilitatem duarum radicum Ovali, Nodo, Cuf-
pide & Punfto conjugate privatur. Eodem fenfu
Parabolam quoq; convergentem^ dtvergentem^ attri-
bm contrarm ffieditum^ cruciformem^ nodatam^ cuj^
fidatanij funtHatam &^ furam nominabimus.

In cal'u primo li terminus a x' afnrmativus eft Fi- ^J^^-
gura erit Hyperbola triplex cum fex cruribus Hy- redimciJtJ^& '^
perbolicis quae juxta tres Aiymptotos quarum nulioe ^j.'^ ^'"'^^ ^-
lunt parallelde in infinitum progrediuntur,binae juxta ^"'^"^''''
unamquamq; in plagas contrarias. Et hoe Afymp-
toti ft terminus bxx non deeft le mutuo lecabunt
in tribus punftis triangulum (Dd'^'^J inter le con-
tinentes, fin terminus bxx deeft convergent omnes
ad idem pundum. In priori cafu cape AD =
-J, & Ad = Ac^ = Tj^, ac junge Dd, D<^, & erunt
AD, Dd, Dci^tres Afym.ptoti. In pofteriori due
ordinatam quamvis BC, & in ea utrmq; produ61:a
cape hine inde B F & B f fibi mutuo squales oc
in ea ratione ad A B quam habet /6. ad a, jungeq;
AF, Af, & erunt AB, AF, Af tres Aiympoti.
Hanc autem Hyperbolam vocamus redundantem
quia numero crurum Hyperbolicorum Seftiones Co-
nicas iuperat.

In Hyperbola omni redundante fi neq; terminus 7,¥^'' „
e y detit neq; ht b b - 4 a c aequale + a e // a curva nwVprhoU diametUs
lam habebit diametrum, fin eorum alterutrum ac« ^'"Z''" ''''"^""^

u cidat

cidat curva habebit unicam diametrum, & tres fi

utrumque. Diameter autem feinper tranfit per in-

terieftioHem duarum Afymptoton & bilecat redtas

omnes qux ad Afymptotos illas utrinq; terminantur

& parallels iunt & Atymptoto tertice. Eftq; abicifla

AB diameter Figurae quoties terminus ey deert.

Diametrum vero ablblute didtam hie & in fequen-

tibus in vulgari fignificatu iifurpo, nempe pro ab-

fciffa qune paiTim habet ordinatas binas a:quales ad

idem pundum hinc inde infiftentes.

XV. Si Hyperbola redundans nulla m habet diamietrum

'vem!£dJtes qui^rantur ^quationis hujus ax-'H-b x'-j- cxx+dx

cjUAdiametrode- -\-\ = o radices quatuor feu valores ipfius x. Ex

tTTZr^ T ^unto A P, A ^ , A TT , A p. Erigantur ordinate

ros tnanguium VTy -stt, ttT, p t, 6i hx tangent Curvam m punccis

caj>ie?itej. totidem T, T, T , t, & tangendo dabunt iimites Gur-

vx per quos Ipecies ejus innotefcet.
%. 1,2. Nam ii radices omnes AP, A% A?!-^ Ap Iunt

reales, ejufdem figni & inxquales, Curva conliat ex
tribus Hyperbolis , ( infcripta circumfcripta & am-
bigena ) cum Ovali. Hyperbolarum una jacet vcr-
fus D, altera verfus d, tertia verfus ^^^ & Ovalis
iemper jacet intra triangulam Dd"', atq; etiam in-
ter medios Iimites i & t ^ in quibus utiq; tangitur
ab ordinatis ifi & ^t. Et ha:c eft fpecies prima.
i§. 3, 4- Si e radicibus duae maximae A vr, A p, vel dua^ mi-

nimae AP, A^ sequantur inter fe, & ejufdem funt
figni cum alteris duobus, Ovalis & Hyperbola cir-
cumfcripta libi inxicem junguntur coeuntibus earum
pundis contatSusI & t vel T & t, & crura Hyper-
bolae fefe decuffando in Ovalem continuantur, hgu-
ram nodatam efficientia. Quae fpecies eft fecunda.

C 1+7 ]

Si e radicibus tres maxims A/, At, A tb-, vel tres Fig. 5, <y;
minimae A t, a w-, a P squentur inter fe, Nodus in
cuffidem acutiffimum convertetur. Nam crura duo
Hyperbolae circumlcripti^ ibi in angulo cTontadtus
concurrent & non ultra producentur. Et ha?c eft
fpecies tertia.

Si e radicibus dua: media- A^ & At g^quentur in- f'>- 7.
ter fe, punfta conta6tus t & 7 coincidunt, & propte-
rea Ovalis interjeda in punftum evanuit, & conftat
figura ex tribus Hyperbolis, infcripta, circumfcripta
& ambigena cum -pun^lo conjugate. Quae eft fpecies
quarta.

Si duae ex radicibus funt impoffibiles & reliqu3e^'i-7,Sji3,i4^
duse inasquales & ejufdem figni ( nam figna contraria
habere nequeunt,) fur^ habebuntur Hyperbolae tres
fine Ovali vel Nodo vel cufpide vel pundlo conju-
gato, & hae Hyperbolae vel ad latera trianguli ab
Afymptotis comprehenli vel ad angulos ejus jacebunt
& perinde fpeciem vel quintam vel fextam confti-
tuent.

Si e radicibus duae funt aequales Sc alterae duaei^*^- 9,10,15,15.
vel impoftibiles funt vel reales cum fignis quae a fig-
nis cEqualium radicum diverfa funt, figura crucifor-
mis habebitur, nempe duae ex Hyperbolis feinvicem
decuffabunt idq; vel ad verticem trianguli ab A-
fymptotis comprehenfi, vel ad ejus bafem. Quae
duae fpecies funt feptima & o6l:ava.

Si deniq; radices omnes funt impoftibiles vel fi%. 11,12;
omnes funt reales Sc insquales & earum duae funt
affirmative & alterae duae negativae, tunc duae habe-
buntur Hyperbolae ad angulos oppofitos duarum

U u a Afymp-

[1+8]

Afymptoton cum Hyperbola mgumeii circa Afymp-
toton tertiam. Quae ipccies eft nona.

Et hi iunt omnes radicum cafus poffibiles. Nam

ii dua^ radices Hint squales inter ie, & alis duas funt

etiam inter fe cequales, Figura evadet Seclio Conica

cum iinea reda.

xvi. Si Hyperbola redundans liabet unicam tantuin

HyferboUduo- Diametrum fit ejus Diameter Ablcifla AB, & a?qua-

tescvmumcatan-t\oms\\u]\is ax^-j- D xx-|- cx-i-d = o qusre trcs ra-

turn Diamelro. ^[(-^^ fg^ ValorCS X.

f^. 17. Si radices illae funt omnes reales & ejufdem iigni,

Figura conftabit ex Ovali intra triangulum D d o^ ja-
cente & tribus Hyperbolis ad angulos ejus, nempe
circumfcripta ad angulum D & infcriptis duabus ad
angulos d Sc o^ Et haec eft fpecies decima.

f'2- 18. Si radices duae majores funt a^quales 8c tertia ejuf-

dem figni, crura Hyperbola jacentis verfus D {t^ic
decuffabunt in (onnd. Afodi propter contadum Ova-
lis. Qu£e fpecies eft undecima.

F'£- 19' Si tres radices funt a:quales, Hyperbola ifta fit

cujpdata {ine Ovali. Quce fpecies eft duodecima.

-%• 20. Si radices duce minores funt a^quales & tertia ejuf-

dem figni, Ovalis in funHum evanuit. Qus fpecies
eft decima tertia. In fpeciebus quatuor noviffimis
Hyperbola quaj jacet verfus D Afymptotos in (inu
fuo ampleditur, reliquai duas in finu Afymptoton
jacent.

pi£' 2.0, Si duae ex radicibus funt impoflibiles habebuntur tres

jfi 22.' Hyperbolae furce fine Ovali decuffatione vel cufpide.

Fig. 23. Et hujus cafus fpecies funt quatuor, nempe decima

quarta fi Hyperbola circumfcripta jacet verfus D &

decima

decima quinta C\ Hyperbola inicrlpta jacet verilis D,
decima lexta fi Hyperbola circumicripta jacet Tub
bafid"' triaiiguli Dd«^, & decima leptima ii Hyper-
bola infcripta jacet Tub eadem ball.

Si du22 radices funt cequales & tertia ligni diverii^'^^-H-
figura cnt crtw if ormis. Nempe du3c ex tribus Hy- "" ^'
perbolis ieinvicem decuflabuiit idq; vel ad verticein
trianguli ab Aiymptotis comprehenli vel ad ejus ba-
lem. Qux dua; Ipecies funt decima octava 3c decima
nona.

Si duae radices funt iiwequales & ejufdem ligni &
tertia eft figni diverii, duffi habebuntur Hyperbolas
in oppoiitis angulis duarum afymptoton cum Con^
choidalt intermedia. Conchoidalis autem vel jace^^-^"
bit ad eafdem partes afymptoti fuae cum triangulo '^' ~
ab aiymptotis conftituto, vel ad partes contrarias ;
& hi duo cafus conftituunt fpeciem vigefimam & vi-
geiimam primam.

Hyperbola redundans quce habet tres diametros ^^^/V .
conftat ex tribus Hyperbolis in finubus afymptoton redundZJet cum
jacentibus, idq; vel ad angulos trianguli ab afympto- tribmDiamstris.
tis comprehenli vel ad ejus latera. Cafus prior dat jrj; 29!
fpeciem vigefimam fecundamjSc pofterior i'peciem vi*
gefimam tertia m.

Si tres aiymptoti in pundo communi fe mutuo xviii.
decuffant, vertuntur fpecies quinta & fexta in vige- ^,^^/5ti^«
fimam quartam , feptima & odava in vigefimam cum Afvm^mis
quintam, & nona in vigefimam fextam ubi Ansuinea "''^'^ ^£f'^'»«-

*■ ;, r c o • • r "^ Vunctiim coil"

non traniit per concurlum aiymptoton, oc m vigen- vergemihts. ' '
mam feptimam ubi tranfit per concurfum ilium, quo ^^- 3o«
cafu termini b ac d defunt, & concu:fus afympto-f-|'32'
ton eft centrum figurce ab omnibus ejus partibusf5-33»

oppofitis

oppofitis squaliter diftans. Et hx quatuor fpecles
-Diametrum non habent.
F(f. 34. Vertuntur etiam fpecies declma quarta ac declma

pf; !^" fexta in vigefimam odavam, decima quinta ac de-

i^/J. 37. ciina feptima in vigefimam nonam, decima o61:ava

& decima nona in tricefimam, & vigefima cum vige-
(ima prima in triceiimam primam. Et hx fpecies
unicam habent diametrum.
j7^. 38. Ac deniq; fpecies vigefima fecunda & vigefima

tertia vertuntur in fpeciem tricefimam fecundam cu-
jus tres funt Diametri per concurfum afymptoton
tranfeuntes. Quae omnes! converfioncs facillime in-
telliguntur faciendo ut triangulum ab afymptotis
comprehenfum diminuatur donee in punCtum eva-
nefcat.
xix! Si in primo asquationum cafu terminus a x' ne-

de^Eiiv^^j^amt g^^^vus eft, Figura erit Hyberbola defeftiva unicam
trtim non hp.hsn- habcns afymptoton & duo tantum crura Hyperbo'
^''^ lica juxta afymptoton illam in plagas contrarias in-

finite progredientia. Et afymptotos ilia eft Ordi-
nata prima & principalis A G. Si terminus e y non
deeft figura nullam habebit Diametrum, fi deeft ha-
bebit unicam. In priori cafu fpecies fie enume-
rantur.
Tig. 35. Si cequationis hujus a x* = b x'^- c x x -J- d x -1- ; e e,

radices omnes At, A P, A/-, A~, funt reales & in-
oequales, Figura erit Hyperbola anguinea afympto-
ton fiexu contrario amplexa, -cum Ovali conjugata.
Q.U3S fpecies eft tricefima tertia.
^>.40. Si radices duse medix AP &: A^ cequentur inter

fe, Ovalis & Anguinea junguntur fefe decuffantes
in forma Modi. Qua: eft ipecies tricefima quarta.

[150

Si tres radices funt ^quales, Nodus vertetur in ^'i?-4T'
cufpdem acutiffimum in vertice anguinese. Et hasc
eft fpecies tricesima quinta.

Si e tribus radicibus ejufdem ligni dux maximse ^'i- 43-
A/Sc Air fibi mutuo sequantur, Ovalis in funcium
evanuit. Quae fpecies eft triceiima fexta.

Si radices dus quscvis imaginaris funt, Tola ma-
nebit Anguinea fm'a fine Ovali, decuflatione, cuf-
pide vel pund:o conjugate. Si Anguinea ilia noniv>.42:
traniit per pundum A fpecies eft triceiima feptima,
fin traniit per pundum illud A ( id quod contingit -'^^^ 43-
ubi termini b ac d deiunt,) pundum illud A erit
centrum figurse redas omnes per ipfum dudas &
ad Curvam utrinq; terminatas bilecans. Et haec
eft fpecies triceiima odava.

In altero cafu ubi terminus ey deeft Sc propterea XX.
figura Diametrum habet. fi aequationis huius ax', ^yprboUfef-
= bxX'T-cx-4-d radices omnes A i, At, At, iunt M?>etn,m h^h^-
reales, inaequalcs & ejufdem figni, figura erit Hyper- y.^;
bola Conchoidalis cum OW? ad convexitatem. (lux ''^''^^'
eft fpecies tricefima nona.

Si duae radices funt inaequales & ejufdem figni & %• 44-
tertia eft figni contrarii, OvaUs jacebit ad concavi-
tatem Conchoidalis. Eftq; fpecies quadragefima.

Si radices duas minores AT, At, funt acquales /■/>.. v6-.
&. tertia At eft ejufdem figni, Ovalis & Conchoi-
dalis j^ngentur iKe decuffando in modum Modi.
Quas fpecies eft quadragefima prima.

Si tres radices funt aequales, Nodus mutabitur in -fi>. 47I
Cufpdem & iigura erit Ctjfois Veterum. Et haec eft
fpecies quadragefima fecunda,.

[152] •

f/j. 49. Si radices duce majorcs fiint acqiialcs, Sc tcrtia eft

ejufdem figni,Conchoidalis habebit fimHum conju-
gatuiii ad convexitatem luam, eftq; Ipecies quadra-
gelima teitia.
Fig. 49. Si radices dux funt gequales & tertia eft ligni con-

trarii Conchoidalis habebit funSlum conjugatum
ad concavitatem luam, eftq; Ipecies quadragefima
quarta,
P/v. 48,49. Si radices duse funt impoftibiles habebitur Con-

choidalis fiira line Ovali , Nodo, Culpide vel
pundo conjugato. Quoe Ipecies eft quadragefima
quinta.
XXI. -Siquando in primo aequationum cafu terminus ax'

^^Tar£i(cl' ^^^^ & terminus bxx non deeft, Figura erit Hy-
Diametrum non perbola Parabolica duo habens crura HyperboHca ad
hahentes, uuam Alymptotou SAG& duoParaboHca in pla-

gam unarn & eandem conver$i;entia. Si terminus
ey non deeft figura nullam habebit diametrum, fin
deeft habebit unicam. In priori cafu Ipecies funt
haf.
.f;V. 50^ Si tres radices AP, A -or, At squationis hujus

bx^-jf-cx Hdx-j-; ee=o funt inaequales & ejufdem
figni, figura conftabit c%.Ovali & ahis duabus Curvis
qux partim Hyperbolic:^ funt & partim Parabolical.
Nempe crura Parabolica continuo duftu junguntur
cruribus Hyperbolicis fibi proximis. Et hsc eft
fpecies quadregefima fexta.
-F/f. 51.. Si radices dua minores funt aequales Sc tertia eft

ejufdem figni, Ovalis & una Curvarum illarum
Hyperboio-Parabolicarum junguntur & fe decuffant
in formam JSfodi. Quce Ipecies eft quadragefima
feptima.

[153]

Si tres radices funt cequales, Nodus ille in Cuf- Fig. 52.
pidem vertitur. Eftq; fpecies quadragefima o<3:ava.

Si radices duae inajores iiint aequales & tertia eft p^z- S3-
ejufdem figni, Ovalis in. funHum conjugatum eva-
nuit. Qucs fpecies eft quadragefima nona.

Si duas radices funt impofiibiles, manebunt fur^e^'i- 53,54-
iWx du32 curvac Hyperbolo'parabolicae fine Ovali,
decuffatione, cufpide vel pun6to conjugate, & fpe-
ciem quinquagefimam conftituent.

Si radices diiae funt cequales & tertia eft figni con^ ^''' 55-
trarii, Curvce illce hyperbolo-parabolica! junguntur
fefe decufl'ando in morem crucis. Eftq; fpecies quin-
quagefima prima.

Si radices duse funt insequales & ejufdem figni & ^'S- 5^-
tertia eft figni contrarii, figura evadet Hyperbola
anguinea circa Afymptoton AG, cum Parabola con-
jugata. Et ha?c eft fpecies quinquagefima fecunda.

In altero cafu ubi terminus ey deeft & figura ^^"•
Diametrum habet, fi du^E radices squationis hujus tuor^ZrlhoET
b xx-|- ex -|-d = o funt impofiibiles, duae habentur -^""«">'«'» ^'«'-
figuras hyperbolo-parabolica^ a Diametro A B hinc FiTin,
inde aequaliter diftantes. Quae fpecies eft* quinqua-
gefima tertia.

Si asquationis illius radices duae funt impofiibiles, %• s8.
Figurse hyperbolo-parabolicse junguntur Mq de-
cuflantes in morem crucis, & fpeciem quinquagefi-
mam quartam conftituunt.

Si radices illae funt inaequales Sc ejufdem figni, "ha- ^'i- Sp-
betur Hyperbola Conchoidalis cum Parabola ex
eodem latere Afymptoti. Eftq; fpecies quinquage-
fima quinta.

X X Si

[I54-]

ri£.6o: Si radices illae funt figni contrarii, habetur Con-

choidalis cum Parabola ad alteras partes Afymptoti.

Quae fpecies eft quinquagefima fexta.

XXIII. [Siquando in primo sequationum calu terminus

pe&TrmTif^c' uterq;ax' &bxx deeft, iigura erit Hyperbolifmus

Ma. ^^"" fedionis alicujus Conies. Hyperbolilmum figurae

voco cujus Ordinataproditapplicandocontentumlub

Ordinata figurse illius & reda data ad Ablcifliim com-

munem. Hac ratione linea reda vertitur in hyper-

bolam Conicam, & fedio omnis Conica vertitur in

aliquam figurarum quas hie Hyperbolilmos ledio-

num Conicarum voco. Nam aequatio ad figuras

de quibus animus, nempe xy y-|-ey = cx-|-d, leu

_ et//ee-r4.dx -|- 4 cxx generatur appli-

cando contentum fub Ordinata fedionis Conica?
ej:/^ee-i-4dxH-'4cxx & red:a data m ad curvarum

2 m

Abfciflam communem x. Unde liquet quod figura
genita Hyperboliimus erit Hyperbola?, Ellipieos vel
Parabolte perinde ut terminus ex affirmativus eft
vel negativus vel nuUus.

Hyperbolifmus Hyperbolae tres habet afymptotos
quarum una eft Ordinata prima & principalis A d,
alterae duae funt parallelae Abfciflae A B & ab eadem
hinc inde aequaliter diftant. In Ordinata principali
Ad cape Ad, A='^ hinc inde a^quales quantitati //c
& per pundta d ac ^ age dg, <^ 7 Afymptotos Ab-
fciflae A B parallelas.

Ubi terminus ey non deeft figura nullam ha-
bet diametrum. In hoc cafu fi cequationis hujus
c X X -j- d X -j- ^e e^o radices duce A P, Ap funt reales

[155]

oc ina'quales ( nam aequales effe nequeunt nifi figura %^ tft.
fit Conica fedtio ) figura conftabit ex tribus Hyper-
bolis fibi oppofitis quarum una jacet inter afymp-
totos parallelas & alterae duae jacent extra. Et haec
eft fpecies quinquagelima feptima.

Si radices ills duoe iiint impoffibiles,habentur Hy-
perbolae dus oppolitx extra afymptotos parallelas &
Anguinea hyperbolica intra eafdem. Haec figura
duarum eft Ipecierum. Nam centrum non habet^^-<yiJ
ubi terminus d non deeft ; fed fi terminus ille deeft ''^* ^^^
pundum A eft ejus centrum. Prior fpecies eft quin-
quagelima odava, pofterior quinquagefima nona.

Quod fi terminus ey deeft, figura conftabit exHf. 54;
tribus hyperbolis oppofitis quarum una jacet inter
afymptotos parallelas 8c alterae duae jacent extra ut
in fpecie quinquagefima quarta, & praeterea diame-
trum habet quae eft abfcilfa A B. Et haec eft fpecies^
fexagefima.

Hyperbolifmus EUipfeos per banc sequationem de- xxiv:
finitur X y y -1- e y = c x-1- d, & unicam habet afymp- t^^,^' ^yp^-'I'ol'A
toton quae eit Ordmata prmcipahs A d. bi terminus Fig. 65.
ey non deeft, figura eft Hyperbola anguinea fine dia-
metro atq; etiam fine centro fi terminus d non deeft.
Qu2 fpeeies eft fexagefima prima.

At fi terminus d deeft, figura habet centrum fine %• 66.
diametro & centrum ejus eft puniftum A. Species
vero eft fexagefima fecunda.

Et fi terminus ey deeft & terminus d non deeft, -f/i^- <^7-
figura eft Conchoidalis ad afymptoton A G, habetq;
diametrum fine centro, & diameter ejus eft Abfcifla
AB. Quae fpecies eft fexagefima tertia.

Xx 1 Hyper-^

Ci5<^]

XXV.

f/>. 68.

Fl£. 6$.

XXVL

Tridens.

Hyperbolifmus Parabolse per hanc aeqiiationem

//r^^kSl'^^^^^^it^i'^yy-^'^y^^^ ^ duashabet aiymptotos,
Abfciflam AB & Ordinatam primam & principalcm
AG. Hyperbolge vero in hac figura iuntdui?, non
in aiymptoton angulis oppcfitis fed in angulis qui
funt deinceps jacentes, idq; ad utrumq; latus ab-
fdffccAB, &. vel fine diametro fi terminus ey ha-
betur, vel cum diametro fi terminus ille deeft. Quae
du9B Ipecies funt fexagefima quarta & fexagefima
quinta.

In fecundo aequationum cafu habebatur squatia
xy = ax'-|-bxx-|-cx-|-d. Et figura in hoc cafu
habet quatuor crura infinita quorum duo funt hy-
perbolica circa afymptoton AG in contrarias partes
tendentia & duo Parabolica convergentia & cum
prioribus fpeciem Tridentis fere eiformantia. Eftq;
haec Figura Parabola ilia per quam Cartefius aequa-
tiones fex dimenfionum conftruxit. Haec eft igitur
fpecies fexagefima fexta.

In tertio cafu aequatio erat yy = ax'-l"bxx-|-cx

FaraboUtjuin- j^^^ & Parabolum defignat cujus crura divergunt

^ue ivergantis. ^ inyicem & in contrarias partes infinite progre-

diuntur. Abfcifla AB eft ejus diameter & fpecies ejus

funt quinq; fequentes. •

Siaequationisax^-|-bx^-l-cx-l-d = o radices om-
nes At , AT, At funt reales & inaequales, figura eft
Parabola divergens campaniformis cum Ovdi ad
verticem. Et fpecies eft fexagefima feptima.

Si radices duae funt aequales, Parabola prodit vel
nodata contingendo Ovalem, vel pun^ata ob Ovalem
infinite parvam. Quae duae fpecies funt fexagefima
0(itava & fexasefima nona.

F;>. 16.

XXVIl.

Fig. noy 11.

Fig. 72.
F^£' 73-

[157]

Si tres radices funt asquales Parabola erit cufpi- -%. 75.
data in vertice. Et haec eft Parabola Neiliana qucB
vulgo lemicubica dicitur.

Si radices duae lunt impoffibiles, habetnr Parabola F/>. 73, 74.
pu7'a campanitbnnis ipeciem ieptuagciimam primam
conftituens.

In quarto cafu aequato erat y = ax'|-bxx+cx xxviil.
+ d, & base sequatio Parabolam ilkm IVallifianam ^^J""^''" '"^'''''
defignat quae crura habet contraria & cubica di- '^'
CL Iblet. Et fie fpecies omnino funt leptuaginta
dua?.

Si in planum Infinitum a pundo lucido illumina' ^^^^•
tum umbras tigurarum projiciantur, umbrae lectio- j.„;„p^^\>;7j^^^.
numConicarum Temper erunt fediones Conicae, eas
Curvarum fecundi generis Temper erunt Curvae fe-
cundi generis, eae curvarum tertii generis Temper
erunt Curvae tertii generis, & fie deinceps in infini-
tum. Et quemadmodum Circulus umbram proji-
ciendo generat Tediones omnes conicas, fie Parabolas
quinq; divergentes umbris Tuis generant & exhi-
bent alias omnes Tecundi generis curvas , Sc fie
Curvae quaedam fimpliciores aliorum generum inve*
niri poffunt quae alias omnes eorundem generum
curvas umbris Tuis* a pun6to lucido in planum pro*
jedtis formabunt.

Diximus Curvas fecundi generis a linea re<3:a in xxx.
pundis tribus Tecari poflfe. Horum duo nonnun- [iaZupii""^""^
quam coincidunt. Ut cum re^ta per Ovalem infi*
nite parvam tranfit vel per concurfum duarum par-
tium Curvae fe mutuo lecantium vel in cufpidem
coeuntium ducitur, Et fiquando re^tae omnes in

plagara

ifcia.

[158]..

plagam cruris alicujus infiniti tendentes Curvam
in iinico tantum pundo lecant ( ut fit in ordinatis
Parabolas Cartefiimae & Parabolae cubicae, nee non in
redis Abiciffae Hyperbolifmorum Hyberbolae Sc Para-
bolae parallelis ) concipiendum eft quod red:ae illae
per alia duo Curvse punfta ad infinitam diftan-
tiam lita ( ut ita dicam ) tranleunt. Hujufinodi
interfediones duas coincidentes five ad finitam
fint diftantiam five ad infinitam, vocabimus pun-
6tum duplex. Curvge autem quas habent pun-
dum duplex deicribi pofiiuit per fequentia Theo-
remata.

XXXI.

Theoremata de , 3^ auguli duo masnitudiue dati PAD, PBD circa
fcriptione orga- polos pofitione datos A, B rotentur, «& eorum crura
A P, B P concurfii iuo P percurrant lineam redam ;
crura duo reliqua A D, B D concurfu fuo D delcri-
bent fe^tionem Conicam per polos A, B tranfeun-
tern : prceterquam ubi linea ilia red:a tranfit per po-
lorum alterutrum A vel B, vel anguli BAD, ABD
fimul evanefizunt, quibus in cafibus pundtum D de-
fcribet lineam redam.

0.. Si crura prima A P, B P concurfu fuo P
percurrant fedionem Conicam per polum alter-
utrum A tranfeuntem, crura duo reliqua A D, B D
concurfu fuo D defcribent Curvam fecundi gene-
ris per polum alterum B tranfeuntem & pun-
dum duplex habentem in polo primo A per quern
fedio Conica tranfit : praeterquam ubi anguli
BAD', ABD fimul evanefcunt, quo cafu pun-

dtum

nica.

[159]

£l:um D defcribet aliam fedionem Conicani per po-
lum A tranfeuntem.

^. At fi fedio Conica quam punftum P perciir-
rit tranfeat per neutrum polorum A, B, punduin
D defcribet curvam fecundi vel tertii generis pun-
6tum duplex habentem. Et pundum illud duplex
in concurfu crurum defcribentium, A D, B D in-
venietur ubi anguli BAP, A B P fimul evanefcunt.
Curva autem delcripta fecundi erit generis ii an-
guli BAD, A B D limul evanefcunt, alias erit ter-
tii generis & alia duo habebit pun^ta duplicia in
polis A & B.

Jam feftio Conica determinatur ex datis ejus xxxii.
punais quinq; & per cadem fie deferibi poteft. ,i^^^Tfi^
Dentur ejus punda quinq; A, B, C, D, E. Jun- f'o per data ^um-
gantur eorum tria quaevis A, B, C & trianguli A BC ^'"^"''^'''
rotentur anguli duo quivis CAB, C B A circa ver-
tices fuos A & B, & ubi crurum AC, BC interfedlio
C fucceffive applicatur ad puntla duo reliqua D, E,
incidat interfedio crurum reliquorum A B & B A
in punda P & Q. Agatur & infinite producatur
redaPQ, & anguli mobiles ita rotentur ut inter*
fedio crurum AB, BA percurrat redam PQ, &:
crurum reliquorum interfedio C defcribet propofi-
tam fedionem Conicam per Theorema primum.

XXXIII.

Curvge omnes fecundi generis pundum duplex cmdigemrispun'
habentes determinantur ex datis earum pundis ^«'». ^y^-^ ^^-
feptem, quorum unum eft pundum illud duplex, J,''p"'^if"J^p

& tern puniia.

[i6o]

& per eadem punda (ic deicribl poffunt. Dentur
Curvx defcribends pun6la quoelibet leptem A, B, C,
D, E, F, G quorum A ell pundum duplex. Jun-
gantur pundum A & alia duo qusvis e pundis puta
B& C; & trianguli ABC rotetur tuin angulus
CAB circa verticem fuum A, turn angulorum rell-
quorum alteruter ABC circa verticem Ilium B. Et
ubi crurum AC, BC concurius C llicceffive appli-
catur ad pun£ta quatuor reliqua D, E, F, G incidat
concurius crurum reliquorum A B & B A in pund:a
quatuor P, Q, R, S. Per pun6ta ilia quatuor &
quintum A defcribatur fedio Conica, & anguli pra?-
fati CAB, CBA ita rotentur ut crurum AB, B A
concurfus percurrat fedionem illam Conicam , &
concurfus reliquorum crurum A C, B C defcribet
Curvam propolitam per Theorema iecundum.

Si vice pundi C datur pofitione reda B C qus
Curvam defcribendam tangit in B, lines A D, A P
coincident, & vice anguli D AP habebitur linea redla
circa polum A rotanda.

Si pundum duplex A infinite diftat debebit Reda
ad plagam pundi illius perpetuo dirigi & mptu pa-
rallelo ferri interea dum angulus ABC circa polum
B rotatur.

Defcribi etiam poffunt has curvae paulo aliter per
Theorema tertium, led deicriptionem limpliciorem
pofuiffe fufficit.

Eadem methodo Curvas tertii, quarti & fuperio-
rum generum defcribere licet, non omnes quidem
fed quotquot ratione aliqua commoda per motum
localem defcribi poffunt. Nam curvam aliquam

fecundi

varum.

fecundi vel fuperiorls generis pun61:um duplex non
habentem commode defcribere Problema eft inter
difficiliora numiCrandum.

Curvarum ufus in Geometria eft ut per earum xxxiv.
interleaiones Problemata Iblvantur. Proponatur ^rftfoLZ^lZ
asquatio conftruenda dimenfionum novem x^*-\-hx^fi^'ptio»emCur
-\- c x° -1- d x^ -\' e xH f ^^ -j- g X X -^f h X -(- k = o. llbi " '

b, c, d, }s^c\ fignificant quantitates quafvis datas
lignis fuis 4- & — ' affectas. Aftlimatur aequatio ad
Parabolam cubicam x' = y, & sequatio prior, Icri-
bendo y pro x', evadet y^-| bxyy -|- cyy-|-dxxy
-r e X y -[- m y -|- f x5 -|- g x x -\- h x -j-k = o, oequatio ad
Curvam aliam fecundi generis. Ubi m vel f deefTe
poteft vel pro lubitu aflumi. Et per harum Curva-
rum defcriptiones & interfediones dabuntur radices
fequationis conftruendae. Parabolam cubicam lemel
defcribere fufficit.

Si squatio conftruenda per defectum duorum ter-
minorum ultimorum hx & k reducatur ad feptem
dimenfiones, Curva altera delendo m, habebit pun-
ftum duplex in principio abfciffe, & inde facile de-
fcribi poteft ut fupra.

Si a:quatio conftruenda per defedum tennino-
rum trium ultimorum gxx-|-hx-l-k reducatur ad
fex dimenfiones, Curva altera delendo f evadet
fedio Conica.

Et fi per defectum fex ultimorum terminorum
asquatio conftruenda reducatur ad tres dimenfiones,
incidetur in conftrudionem IValitfianam per Para-
bolam cubicam & lineam reitam.

y y Con-

[162]

Conftrui etiam poffunt sequationes per Hyperbo-
lilmum Parabola cum diametro. Ut li conllruenda
fit hcEC asquatio dimenlionum novem termino penul-
timo carens, a -|- c x x -|- d x' -j- ex* -\- f x + g x^ -| h x^

-j-kx^ -(-1x9 = 0 ; aiTumatur aequatio ad Hyj^erbolil-
mum ilium xxy= i, & Icribendo y pro ~, oequatio
conftruenda vertetur in hanc ay' -1' c y y -|- d x yy -j- e y
-|- f X y -1- m X X y -|- g-l" h X -1- k x X -|- 1 x' = o, quse cur-
vam fecundi generis delignat cujus defcriptione
Problema folvetur. Et quantitatum m ac g alter-
utra hie deefTe poteft, vel pro lubitu affumi.

Per Parabolam cubicam & Curvas tertii generis
conftruuntur etiam a^quationes omnes dimeniionum
non plulquam duodecim, & per eandem Parabolam
& curvas quarti generis conftruuntur omnes dimen-
iionum non plufquam quindecim, Et fie deinceps iti
infinitum. Et curvae illae tertii quarti & fuperiorum
generum defcribi Temper poffunt inveniendo eorum
pun6ta per Geometriam planam. Ut fi conftruenda
lit sequatio x" * -\- a x'°-|- b x'-j- c x^-|- d x'-j- e x^-|- f x«
-Vgx'* -|- hx^ -\- ixx -1- kx -|- 1 = o , & defcripta
habeatur Parabola Cubica ; fit aequatio ad Pa-
rabolam illarn cubicam x^ = y , & fcribendo y
pro X* oequatio conftruenda vertetur in hanc
y4 -|-axy^ ^(-cxxyy -j-fxxy -|-ixx = o , quas eft
-Vb -i-dx -J-gx -^kx

•+e +h _+i^

aequatio ad Curvam tertii generis cujus defcriptione
Problema folvetur. Defcribi autem poteft htrc Curva
inveniendo ejus pundla per Geometriam planam,prop-
terea quod indeterminata quantitas x non nifi ad
duas dimenfiones afcendit.

TRACTATE

D E

Quadratura Ciirvarum.

C i<J$ 3

INTRODUCTIO.

Qllantltates Mathematicas non ut ex partibus
quam minimis conftantes, fed ut motu conti'
nuo defcriptas hie confidero. Linese delcri-
buntur ac defcribendo generantur noii per appoli-
tionem partium fed per motum continuum pundo-
rum, fuperficies per motum linearum^ folida per
motum fuperficierum, anguli per rotationem late-
rum, tempora per fluxum continuum, & fie in cce-
teris. Ha? Genefes in rerum natura locum vere ha-
bent & in motu corporum quotidie cernuntur. Et
ad hunc modum Veteres ducendo redtas mobiles in
longitudinem re(^arum immobilium genefin docue-
runt re^tangulorum.

Confiderando igitur quod quantitates aequalibus
temporibus crefcentes & crefcendo genitse, pro velo-
citate majori vel minori qua crefcunt ac generantur,
evadunt majores vel minores ; methodum qujerebam

deter =

'detei'minandi quantitates ex velocitalibus motuum
vel incrementoruni quibus generantur \ & has mo-
tuum vel incremcntorum velocitates nominando Flu-
xtones & quantitates genitas nominando Fluentes^ in-
cidi p3.uhtim u4nnis i665&i666in Methodum Flu-
xionum qua hie ulus ium in Quadratura Curvarum.
Fluxiones funt quam proxime ut Fluentium aug-
menta a^qualibus temporis particulis quam minimJs
genita, & ut accurate loquar, funt in prima ratione
augmentorum nafcentium ; exponi autem poiTunt per
lineas quafcunq; quae funt iplis proportionales. Ut
'/g . I . fi arese A B C , A B D G Ordinatis B C , B D fuper

ball A B uniformi cum motu progredientibus defcri-
bantur, harum arearum fluxiones erunt inter fe ut
Ordinate defcribentes BC & BD, & per Ordinatas
illas exponi pofTunt, propterea quod Ordinatce ilia;
funt ut arearum augmenta nafcentia. Progre-
diatur Ordinata BC de loco fuo BC in locum
quemvis novum b c. Compleatur parallelogram-
mum BCEb, ac ducatur reda VTH quae Cur-
vam tangat inC ipfifq; be & B A produdis occur^
rat in T & V : & Abfciffe AB, Ordinate^ BC, &
Lineae Curvx ACc augmenta modo genita erunt
B b, E c & C c ; & in horum augmentorum nafcen-
tium ratione prima funt latera trianguli CET,ideoq;
fluxiones ipfarum AB, BC & AC funt ut trianguli
illiusCET latera CE, ET&CT & per eadera
latera exponi poifunt, vel quod perinde eft per la-^
tera trianguli confimilis VBC.

Eodem arccidit fi fumantur fluxiones in ultima
'ratione partium evanefcentium. Agatur reda Cc
& producatur eadem ad K. Red eat Ordinata be

in locum fuum priorem B C, & coeuntibus pun6fcis
C & c, reda C K coincidet cum tangente C H, 8c
triangulum evanefcens CEc in ultima fua forma
evadet fimile triangulo GET, Scejuslatera evanef-
centia CE, Ec & Cc erunt ultimo inter feut funt
trianguli alterius GET lateraGE, ET&GT, &;
propterea in hac ratione iunt fluxiones linearum A B, .
BG&AG. Si pun6ta G & c parvo quovis inter'
vallo ab invicem diftant re6:a CK parvo intervalio a
tangente GHdiftabit. Utre61:a GK cum tangente-
G H coincidat & rationes ultimse linearum G E, E c &
Gc inveniantur, debent pun6la G & c coire & cm-
nino coincidere. Errores quam minimi^ in rebus,
mathematicis non funt contemnendi.

Simili argumento ficirculus centre B radio BG
defcriptus in longitudinem Abfciffae A B ad angulos
rectos uniform! cum motu ducatur, fluxio folidi ge-
niti A B G erit ut circulus ille generans, & fluxio fu-
perficici ejus erit ut perimeter Girculi illius &
tiuxio lineas curvae A G conjunftim. Nam quo tem-
pore folidum ABG generatur ducendo circulum
ilium in longitudinem Abfcifla? A B, eodem fuper-
ficies ejus generatur ducendo perimetrum circuli il-
lius in longitudinem Gurvae A G.

R^Ba TB circa folum datum P revolvens fecet aliam Fig. 2.
fofitione datam redam ^B : quiffritur p'ofortio fiuxio"
num reSlarura tUarum ^B ^ PJ5. ProgrediatUE
re£ta P B de loco fuo P B in locum novum P b. In
P b capiatur P G ipli P B squalls, & ad AB ducatur
P D fie, ut angulus b P D aequalis lit angulo b B G ;
& ob limilitudinem triangulorum bBG, bPDerit
augmentum Bb ad augmentum Cb ut Pb ad Db=

Redeat

Redeat jam P b in locum fuum priorem P B iit r; ag-
menta ilia evanefcant, & evanetcentium ratio ulti-
ma, id eft ratio ultima Pb ad Db, ea erit quae ell
PB ad D B, exiftente angulo PDB reao, & prop-
terea in hac ratione eft ftuxlo iplius A B ad fiuxionem
ipfiusPB.
^i„-^ 3. ReSia T B circa datum Tohm T rtvolvens fecet

alias duos fofitione datas recHas AB^AE in B )^
jE : qucerttur frofortio fiuxionum reBarum iUarum
AB }f^ AE. Progrediatur reda revel vens P B de
loco fuo P B in locum novum P b ledas A B, A E in
pundis b &e lecantem, & rcdce A£ parallela BC
ducatur ipii Pb occurrens in C, 8c erit Bb ad BC ut
Ab ad Ae, & BC ad Ee ut P B ad PE, & conjundis
rationibus Bb ad Ee ut AbxPB ad AtxPE.
Redeat jam linea Pb in locum fuum priorem PB, &
augmentum evaneicens B b erit ad augmentum eva-
neicens Ee ut ABxPB ad AExPE, ideoq; in
hac ratione eft fluxio redtas A B ad fiuxionem reds
AE.

Hinc ft reda revolvens PB lineas qnafvis Curvas
politione datas tecet in pundis B & E, & re61:ae jam
mobiles AB,AE Curvas illas tangant in Sedionum
pundis B & E : erit fluxio Curva- quam reda, A B
tangit ad fiuxionem Curvae quam reda A E tan^it
ut A B>^P B ad A Ey.P E. Id quod etiam eveniet
fi reda PB Curvam aliquam poiitione datam perpe-
tuo tangat in pundo mobili P.

Fluat quantita^ x uniformiter Jf; mveniendafit fluxio
quantitatis x\ Quo tempore quantitas x fiuendo

•evadit x | o, quantitas x'' evadet x-| o|"^ id eft
per methodum lerierum infinitarum, x^-j nox""'

[ i<59 ]

H-i!^oox"-'-i b'<:. Et augmenta o & nox"-'-| lifoox"'*
-J-JfT^r. funt ad invicem ut i & nx"''-l-Hil:?lox"-2-j- Ifj*^.
Evanefcant jam augmenta ilia, & eorum ratio
ultima erit i ad nx"^'' : ideoq; fiuxio quantitatis
X eft ad fluxionem quantitatis x" ut i adnx"-^

Similibus argumentis per methodum rationum
primarum & ultimarum colligi pofTunt fluxiones li-
nearum feu reftarum feu curvarum in cafibus qui-
bufcunque, ut & iiuxiones fuperficierum, angulo-
rum & aliarum quantitatum. In finitis autem quan-
titatibus Analyfin fie inftituere, & finitarum nafcen-
tium vel evanefcentium rationes primas vel ultimas
inveftigare, confonum eft Geometriae Veterum : Sc
volui oftendere quod in Methodo Fluxionum non
opus fit figuras infinite parvas in Geometriam intro-
ducere. Peragi tamen poteft Analyfis in figuris qui-
bufcunq; feu finitis feu infinite parvis quae figuris
evanefcentibus finguntur fimiles, ut & in figuris quae
pro infinite parvis haberi folent, modo caute pro-
cedas.

Ex Fluxionibus invenire Fluentes Problema dif-
ficilius eft, & folutionis primus gradus cequipollet
Quadraturae Curvarum ; de qua fequentia olim
fcripfi.

Zz D E

[i7o]

TRACTATUS

D E

Quadramra Curvarum.

QUantitates indeterminatas ut motu pcrpetuo-
. creicentes vel decrefcentes, id eft ut fluen-
tes vel defluentes in fequentibus confidero,delignoq;
literis z, y, x, v, & earum fluxiones leu celeritates

• • • •

crefcendi noto iifdem literis pundtatis z, y, x, v.
Sunt & harum fluxionum fluxiones feu mutationes
magis aut minus celeres quas ipfarum z, y, x, v
fluxiones fecundas nominare licet & fic dignare

z, y, X, V, & harum fluxiones primas feu ipfarum
z, y, X, V fluxiones tertias fic z, y, x, v, & quartas fic
z, y, X, V. Et quemadmodum z, y, x, v funt flu-
xiones quantitatum z, y, x, v, & hx funt fluxiones

quantitatum z, y, x, v & hae funt fluxiones quantita^
turn primarum z, y, x, v : fic hoe quantitates confide-
tari pofTunt ut fluxiones aliarum quas fic defignabo,

'■' // // ;/

z, y, X, V, & hx ut fluxlones aliarum z, y, x, v, &

hx ut nuxiones aliarum z, y, x, v. Detrgnant igitur

z, z, z, z, z, z, z, z iS''<r. feriem quantitatum quarum
qucelibet pofterior eft fluxio prscedentis & quaelibet
prior eft tiuens quantitas fluxionem habens fubfe-

quentem. Similis eft feries ^az — zz, A^az — zz,

r'az — zz , f^cLz — zz , /^az — zz , A^az — zz , ut 8c

- . az4-z^ az-l-z^ az-l-z^ az-4-z^ az-4--z'
leries " '> ' j j

a — z a — z a — z a — z a — z

9
SZ— i~Z^

J... . Et notandum eft quod quantitas quaelibet

a — z
prior in his feriebus eft ut area figurae curviliniae
cujus ordinatim applicata redtangula eft quantitas

pofterior & abfciffa eft z : uti A^az — zz area curvae

<:ujus ordinata eft A^az — zz & abfcifla z. Quo au-
tern fpe^tant hsc omnia patebit in Propolitionibus
quae fequuntur.

2z a J^ROR

[172]

PROP. I. PROB. I.

^ata icquatione quotcunc[y Jluentes quant hates invol-
vente^ invenire fiuxiones.

Solutio.

Multiplicetur omnis asquationls terminus per In-
dicem dignitatis quantitatis cujufq; fluentis quam
involvit, & in fingulis muitiplicationibus mutetur
dignitatis latus in fluxionem fuam, & aggrega-
tum fa(Sorum Dinnium fub propriis fignis erit
aequatio nova.

ExpUcatio.

Sunto a, b, c, d b'r. quantitates determinatae &
immutabiles, Sc proponatur aequatio quaevis quan-
titates fluentes z, y, x l5fc\ involvens, uti x^ — x y y
-f- a a z — b' = o. Multiplicentur termini primo per
indices dignitatum x, & in iingulis muitiplicationi-
bus pro dignitatis latere, feu x unius dimenfionis,

fcribatur X5& fumma faftorum erit 3 x x' — x y y .Idem
fiat in y & prodibit — ^x y y. Idem fiat in z & pro-
dibit a a z. Ponatur fumma faftorum aequalis ni-
hilo, & habebitur sequatio gxx^ — xyy — 'xyy

-\-3. a z — o. Dico quod hac oequatione definitur re-
latio fluxionum.

"De-

[173]
Demonftratio,

Nam fit o quantitas admodum parva & ilinto

oz, oy, ox, quantitatum z, y, x momenta id eft in-
crementa momentanea fynchrona. Et fi quantita-
tes fluentes jam funt z, y & x, hse poft momentum

temporis incrementis fuis oz, oy, ox auftse, evadent

• • •

z-^-oz, y-l-oy, x-|-ox, qucE in asquatione prima pro
z, y & X j^-ripta! dant aequationem x^ -j-^xxox

• • • • • • •

-1- 5X00XX -)- o3x3 — xyy — oxyy — ^xoyy — -xooyy

■ • • • • •

— xooyy — xo3yy-^-aaz-|-aaoz — bg = o. Subducatur

asquatio prior, & reliduum divifum per o erit ^xxz

-]-3xxox-(-'x5oo — xyy — 2xyy — 2xoyy —xoyy— xooyy
-i-aaz = o. Minuatur quantitas o in infinitumyx ncg-

ledis terminis evanefcentibus reftabit ^xx^ — xyy

— 2xyy4-aaz = o. Q. E. D.

ExpUcatio plenior.

Ad eundem modum fi squatio effet X3 — xyy

— — — — '

-)-aa f^ax — yy — b9 = o, produceretur ^x^x — xyy
'■ — 2xyy-|-aar^ax — yy = o- Ubi fi fluxionem/Ax — yy
tollere velis, pone Kax — yy = z, &: erit ax — yy = z^

[174]

Sc (.per hanc Propofitionem ) ax — ^yy = ^/z feu

a =?r, hofc eft -^-^ = ^ax — yy . Et

^z ^/^ax — yy

. , • • • , a'x — laayy
inde 3x'x — xyy — =^xyy-j ~=Q

^f/^x — yy
Et per operationem repetitam pergitur ad fluxio-
nes lecundas, tertias & ifequentes. Sit aequatio
zy3 — z4-)-a* = o, &: fiet per operationem primain

• • • • • ■ •

zy^-^-t^zyy^ — 4zz5 = o, per fecundats zy^-j-6zyy2
^-3zyy2-|-6zy^y — ^zz? — i'2z2z^ = o, per tertiam

zy^ + 9zyy' + 9^yY^ + i^zy^y + 3zyyM^. ^Szyyy

• • • «• • «

-|^6zy^ — 4.ZZ3 — 36ZZZ2 — a4.z3z = o.

Ubi vero iic pergitur ad fluxiones fecundas, ter-
tias & iequentes, convenit quantitatem aliquam ut
uniformiter liuentem confiderarc,& pro ejus fluxione
prima unitatem Icribere, pro fecunda vero & fe-
quentibus nihil. Sit aequatio zy^ — 7.* ^--: a4 = o, ut
.lupra^ & fluat z uniformiter, fitq; ejus fluxio unitas,

& fiet per operationem primam y^ -[- ^zyy ^ — 4Z3 = o,

per fecundam 6yy^ -\-^ ^zyy^ -\- 6zy^y — 1 2z^ = o,

.per tertiam 9yyM-iSy^y+3zyy'+i8zyyy-l-6zy3

-— 24.Z = 0.

[i7$l -

Th hujus autem generis ^equationibus condpieii-
dum eft quod iiuxiones in fingulis terminis fint ejuf-

dem ordinis, id eft vel omnes primi ordinis y, z,

vel omnes fecundi y, y^, yz, z% vel omnes tertii

• • •

y-> yy? y^i yS y^^-> y^^ '^^ ^^- ^t ubi res aliter fe
habet complendus eft ordo per fubintelledas Iiuxio-
nes quantitatis uniformiter fluentis. Sic aequatio

noviffima complendo ordinem tertium fit ^zyy^

+ 1 8zy^y+ 3Zyy^^- 1 8zyyy-1.6zy3— ^^zz^ = o.

PROP. IL PROB. IL

Jnvenire Curva^ quce qmdrari pjfunt.^

Sit A B C figura invenienda, B C Ordinatim ap- Ftg. ^.
plica ta redangula , 8c AB abfciffa. Producatur
CB ad E ut fit BE=i, & compleatur parallelo-
grammum ABED: & arearum ABC, ABED
fluxiones erunt ut BC & BE. Affumatur igitur
aequatio qusevis qua relatio arearum definiatur, &
inde dabitur relatio ordinatarum BC & BE per
Prop. I. Q. E. I.

Hujus rei exempla habentur in Propofitionibus
duabus fequentibus.

PROP,

[176']

PROP. III. THEOR. I,

Si pro abfciffa A B & area AE feu ABxi pro-
milcuefcribaturZj&lipro e -j-fz" -1-gz^" -l-hz^M-j-Scc.
Icribatur R : fit autem area Curvae zsR" erit.
ordinatim applicata BC =

Demonftratio.

Nam fi fit z9R'^=v, erit per Prop, i, ^zz^'Ra

^-AZ^RR^'^ = v. Pro R'^ in primo aequationis ter-
mino Sc z' in fecundo fcribe RR'^'' & zz^', & fiet

• • •

•zR-V'^zR in z^'^ R\j = v. Erat autem R = e -)- fz*
+§z=''+hz3« &c. & inde per Prop. i. fit R =;
Hfzz»-'-]-2Hgzz^«-'-l-j«hzz3«-^-l- &c. quibus fubftitu-
tis & fcripta B E feu i pro z, fiet
•e+;;j-_fz»^:;^gz=H4,,hz3"^-.&c. in z9-R-'=v = BC.
Q. E. D.

PROP.

[177]

PROP. IV. THEOR. II.

Si Curvae abfciffa A B fit z, & fi pro e-]- f z" -(-gz**
-j-^&c. fcribatur R, & pro k-\-\vi-\-'mz^''-\- &c. fcri-
batur S ; fit autem area Curvte z^ R*^ S** : erit or-
dinatim applicata B C =

• y^ • • • • * ^ • « • • • "^ '

.ek+;.fkz«-^4,gkz'.

Demonftratur ad modum Propofitionis fiiperioris.

PROP. V. THEOR. III.

SiCurvse abfcilTa AB fit z, & pro e-l-fz''-l-gz^''
''\- hz3" -f- &c. fcribatur R : fit autem ordinatim ap-
plicata z^-'R'^" in a -^-bz" ^cz^" -l-dz5''-l- &c. & po-

natur ^=^r. r-j-'^^s. s-(-'^ = t. t-l-^ = v.&c. erit area

z jv in — -\- -zw -|- — — z'^»-\-- zi»

re rH-i,e" r4-2,e r'^+TJe

^_ r3fDzUC=:hB ^^^ _^_ ^^^ ^j^. ^^ g^ ^^ j^^ ^^^

r-H-4,e

Aaa denotant

[;t78]

denotant totas coefficientes datas terminorum fingii-
Jorum in ferie cum fignis iuis-l-Sc — ,nempe A primi

termini coefficientem jil B fecandi coefficientem

^b-^sfA p^ .. J'.' ^ ,7crJB— tgA

.. , ^ tertll rnefflripnf-pm g^

r H- I , e

lie dcincep?.

I, e r -}-2,e

Demonjlratio.
Sunto juxta Propofitionem tertiam,

Curvarum Ordinatse & earundem arese.

I. eeA :|:;„fAz" :j4^g Az^« :j:9^^hAz3''&c. I Az« R\

a etii, eBz" i-fl^f B z^" "ff^Bz?" &c. [ Bze+" R\

3 • • • • • +9TrH,eCz^'':f9;H''fCz3" &c. Cz^-l'^- R\

4. -h9+7«,eDz3»&c. J Dz^ i-3» ra

Et fi fumma oidinatarum ponatur aequalis ordi-
natae a-(-bz«-|-cz'=''-l-dz5"-l- Sec. in z^-'R'^-', fumma
arearum z^R'^ in A-j-Bz»-l-Cz^"-|-Dz3''-|- &c. squa-
liseritarea^ Curvos cujus ifta eft ordinata. ^quen-
tur igitur Ordinatarum termini correlpondentes, &
fiet a=.eA, h^^ih-^^^h, c= .,^lg^-\^' fB

'J- et^jC C &c. & inde 5^ = A. — 1_ — = B.

' 9-l'«>e

C-fgl^, gA-Hr|-A«,fB ^ T7. r J • • • r

~- e-l-2«,e- — — = C Jbt he demceps m mfi-

C 179 ]

iiituin . Pone jam J = r . r -|- ^ = s . s -j- ^ = t &c. &
in area z^R^x A+Bz"-j-Cz^''-|~Dz3« &c. fcribe ip-
forum A, B, C, &c. valores inventos & prodibit
leries propoiita. Q. E. D.

Et notandum eft quod Ordinata omnis duobus
modis iu feriem refolvitur. Nam index " vel affir-
mativus eft poteft vel negativus. Proponatur Ordi-
nata — 7^'V^^ . HsEc vel fie fcribi poteft

"^ ,7/1,7 — Iz2--mz4. A

zz^kz— Iz3-l-mz4

z-^xgk — Izzxk — lzz-|-mz^|'^, vel lie zx-l-j-^kz-*
xm-lz"''pkz~^, —i. In cafu priore eft a= 3k.b = o.
c=-l. e=k. f=o. g= -1. h=m. A:=-i. "=i.
6-1=-^. 9=-|=r. s=-i. t=-^. v=o. In
pofteriore eft a=:-l. b^o. c=3k. e=m. f=~l.
g=o. h=i. A=_i. H=-i. 9-1 = 1.9=1. r=-2.
s=— i^. t=— I. v=— ^. Tentandus eft cafus uter-
que. Et (i ferierum alterutra ob terminos tandem
deficientes abrumpitur ac terminatur, habebitur area
Curvae in terminis tinitis. Sic in exempli hujus
priore cafu fcribendo in ferie valores ipforum a, b,
c, e, f, g, h, A, 9, r, s, t, v, termini omnes poft pri-
mum evanefcunt in infinitum & area Cur vce prodit
— ^V ~^""''"^^ Et hac area ob fignum negativum
adjacet abfciffae ultra ordinatam produdae. Nam
area omnis affirmativa adjacet tam abfciftae quam
ordinatse, negativa vero cadit ad contra rias par-
tes ordinatae & adjacet abfciffa? product ae, manente
fcilicet fi2,no Ordinate^. Hoc modo feries alter-
utra & nonnunquam utraque femper terminatur
& finita evadit fi Curva geome trice quadrari po-
teft. At li Curva talem quadraturam non admit-
tit, feries utraq; continuabitur in infinitum, & ea-

A a a 2 rum

[i8o]

rum altera converget & areamdabit approximando,

praeterquam ubi r ( propter aream intinitani ) vel
nihil eft vel numerus integer & negativus, vel ubi ^

cequalis eft unitati. Si ^ minor eft unitate, conver-
get feries in qua index „ affirmativus eft : fml unita

te major eft, converget feries altera. In uno cafu
area adjacet abfciflEE ad ufq; ordinatam duilse, in
altero adjacet abiciftce ultra ordinatam produdce.

Nota infuper quod (i Ordinata contentum eft ftjb
faftore rationali Q. & fadore furdo irreducibili R*,
& fadoris furdi latus R non dividit factorem ratio-
nalem Q; erit a— i =t & R'^-J = R''. Sin fadoris fur-
di latus R dividit fadorem rationalem femel, erit
A-~i = 7r~(- I & R^-' =R'^i'' : ft dividit bis, erit
A~i=7r-|-a & R'^-^ =R''-1'2: ft ter, erit A-i=7r_|_^^
& R^-'=R'^3 : & ficdelnceps.

Si Ordinata eft fradio rationalis irreducibilis cum
Denominatore ex duobus vel pluribus terminis com-
pclito : refolvendus eft denominator in divifores
fuos omnes primos. Et ft divifor fit aliquis cui
nullus alius eft aequalis , Curva quadrari nequit :
Sin duo vel plures fint divifores asquales, rejicien-
dus eft eorum unus, & fi adhuc alii duo vel plures
fint fibi mutuo sequales & prioribus insequales, re-
jiciendus eft etiam eorum unus, & fie in aliis omni-
bus aequalibus fi adhuc plures fint : deinde divifor
qui relinquitur vel contentum fub diviforibus omni-
bus qui relinquuntur, fi plures funt, ponendum eft
pro R, & ejus quadrati reciprocum R'^ pro R''"',prse-
terquam ubi contentum illud eft quadratum vel cu-
bus vel quadrato quadratum,&c. quo cafu ejus latus

ponea^

[i8i]

ponendum eft pro R & poteftatis index 2 vel 5 vel 4.
negative fumptus pro a. & Ordinata ad denomina-
torem R^ vel R' vel R^ vel R' &c. reducenda.

Ut fi ordinata fit ^J5±L^:=^3_ . nnoniam Ir^r

fradio irreduci bills eft & denominatoris divi lores
funt pares, ^ nempe z— i, z— i, z— i & z-|-^a,
z-1-2, rejicio^ magnitudinis utriufque diviforem
unum & reliquorum z— i, z — i , z-|-a conten-
turn z'— 5z-l-a pono pro R & ejus quadrati re-
ciprocum -^^ feu R-^ pro R^-^ Dein Ordina-
tarn ad denominatorem R' feu R'-'^ reduco, & fit

z^-9z'^-l-8z3 ^ . .

^V^,~~r 1 7 , id eft Z3X 8 -9Z-UZ3X 2 - 2z-rzT''

z3-3z-|-a| quad.' ^ 1 ^ 3 i '

Et inde eft a = 8. b=-9. c = o. d=-i, Sec.
e=a. ^=-9. g = o- h=i. ^-i = _2. ;,= _i-.
„=i. 9-1 = 5. 9 = 4-r. 8=^. t = a. v=i. Ethis
in ferie fcriptis prodit area ^t^Vts ' ^^^'"^i^is om-
nibus in tota ferie poft primum evanefcentibus.

Si deniq; Ordinata eft fraftio irreducibilis & ejus
denominator contentum eft fub fadtore rationali Q.
& fadore furdo irreducibili R'', inveniendi funt Ja-
teris R divifores omnes primi, & rejiciendus eft di-
vifor unus magnitudinis cu jufq; ik per divifores
qui reftant , fiqui fint , multiplicandus eft factor
rationalis Q r & fi factum acquale eft lateri R vel
lateris illius poteftati alicui cujus index eft numerus
integer, efto index ille m, & erit a— i = — ^— m, &
Rvi ^ R-,.m. ^^ ^^ Ordinata fit ^^'-^^-r^9^'--r^^^v^ ,

q^-^XX^/CUb. q3-|-qqx — qxx — x 5

quoniam

[l82]

quoniamfadoris furdi latusR feu q^ -l-qqx-qxx—x^
divilbres habet q + x, q-l~x, q — xqui duarumfunt
magnitudinum, rejicio divilbrem unum magnitudi-
nis utriulq; & per divilbrem q+x qui relinquitur
multiplico fadorem rationalem qq — xx. Et quo-
niam failum q^ + qqx — qxx — ^x^ asquale eft la-
teri R,pono m=i . & inde, cum -n fit ], fit a-i =— ^.
Ordinatam igitur reduce ad denominatorem 'R.'l

& fit Z° X 3qM^aq^x-l-8q^xxl-8q'x'~-7qqxC6qx^
X qJ -|- qqx-^qxx — x'hj. Unde eft a = 3 q^ b = 2q^ &c.
e = q3. f=qq&c. 9— 1=0. 9=1=". x = — \- r= i.
s = H. t = '. v = o. Et his in ferie fcriptis prodit

area , -, terminis omnibus in ferie tota

y'cub. aa-J-aax— axx — x' '

poft tertium evanefcentibus.

PROP. VI. THEOR. IV.

Si Curvae abfciffa AB fit z, & fcribantur R pro
e_[-fz«+gz^«+hz3«-l-&c. & S pro k -|- Iz" -j-mz^M
-\-nz3«&c. fit autem ordinatim applicata z^-'R'^-' S^-'
in a-\-bz« -l-cz^^^-dz^" &c. Sc fi terminorum, e, f,
g, h, &c. & k, 1, m, n. &c. redtangula fint.

ek fk gk hk &c.

el fl gl hi &c.

em fm gm hm&c,

en fn gn hn &c.

[183]

Et fi re£tangulorum illorum coefficientes numc-
rales fint refpedive

»9 = r. r -^-7^ = s. s-(-A = t. t -]-A = V. Sec.
s-l-|u=t;'. t-)-*^ — V. v-^-/* = w. w-j-f/^x. occ.
area Curvae erit hsec

-tgk

-T -K— 5fk/\ I, s-l-i,fk-n — t'fl A

z^R'^S'^in — ^4_- ^,fl_l . — L ^

' r-[-i,ek '

^»

rek ' r-[-i,ek r-|-2,ek
-V hkA

LJ — s-l-2,fkp — t'-|-i,fl D — v"fm
« ^ —5-^-2, e l"^ — t'H-i,e m _v"'e n

_j -—- _^ z3« '[i &;c.

r+3. e k

Ubi A denotat termini primi coefficientem dataxn
•li cum %no fuo -1- vel — B coefficientem datam

rek '-^ ' ''

fecundi, C coefficientem datam tertii,&.(ic deinceps.
Terminorum vero, a, b, c, &c. k, 1, m, &c. unus
vel pluresdeeffepofTunt. Demonftratur Propofitio
ad modum praecedentis, & qux ibi notantur hie ob-
tinent. Pergit autemferies talium Propofitionum in
infinitum, & Progreffio feriei manifefta eft.

PROP,

[184.]

PROP. VII. THEOR. V.

Si pro e-^fz«-t-gz2"-l- &c. fcribaturR utfupra, &
in Ciirvse alicujus Ordinata zfliv K»^±r mancant
quantitates dats 9, «, a, e, f, g, &c. & pro o- ac t Icri-
bantur fucceffive numeri quicunq; integri : & fi
detur area unius ex Curvis quce per Ordinatas in-
numeras fie prodeuntes defignantur fi Ordinatae funt
duorum nominum in vinculo radicis, vei li dentur
areoe duarum ex Curvis fi Ordinatoe lunt trium nor
minum in vinculo radicis, vel area trium ex Curvis
fi Ordinatas lunt quatuor nominum in vinculo radi-
cis, & fie deinceps in infinitum : dico quod dabun-
tur ares curvarum omnium. Pro nominibus hie
habeo terminos omnes in vinculo radicis tam de-
ficientes quam plenos quorum indices dignitatum
funt in progreffione arithmetica. Sic ordinata
Va"^ — ax^ -j- x'^ ob terminos duos inter a* & — ax^
deficientes pro quinquinomio haberi debet. At

Va'*-1-X4 binomium eft & Va^-l-x'^ — — trinonium,

cumprogreffio jam per majores differentias proce-
dat. Propofitio vero fie demonftratur.

C ^ S. I.

Sunto Curvarum duarum Ordinatce pz®'^ R'^'' &
qx9 ! «-i ra-i^ ^ ^j.^^ p^ 3. (^g^ exiftente R quanti-

tate trium nominum e-j-fzw-j-gz^*. Et cum per

Prop.

[185]

Prop. in. fit zflR'^ area curvae cujus Ordinata eft
8^-1 J/^"-'! a.,gz'" in zS-'R^-',(ubduc Ordlnatas & areas
priores de area & Ordinata pofteriori, & manebit
!:p t'/^"'| L g^'^i" z^-R^-' Ordinata nova CurvsE,^

— qz"

x9R^ — pA — qB ejufdem area. Pone 9e = p &
flf_j-,^«f =:q Si. Ordinata evadet , J^^ gz^" in z^-'R'^-', &
area z^R'' — seA — gfB — A«fB. Divide utramq; per
flg-j-'AHg, &: aream prodeiintem die C, & aflumpta
utcunq; r, erit r C area Curvae cujus Ordinata eft
i-gfl i 2«-i j^A-i £^- q^^jj ratione ex areis pA &: qB
aream rC Ordinate rz^l'^^-i ^k-i congruentem inve-
nimus, licebit ex areis qB & rC aream quartam
puta sD, ordinatae sz^'l^""R'^'' congruentem invenire,
&i fic deinceps in infinitum. Et par eft ratio pro-
greffionis ab areis B & A in partem contrariam
pergentis. Si terminorum 9, 9 -j-'^", & 9-\-2,,„ aliquis de-
ficit & feriem abrumpit, aflumatur area pA in prin-
cipio progreffionis unius & area qB in principio al-
terius, & ex his duabus areis dabuntur areae omnes
in progreffione utraque. Et contra, ex aliis duabus
areis aflumptis fit regrelTus per analyfin ad areas A
& Bj adeo ut ex duabus datis caeteras omnes den-
tur. Q. E. O. Hie eft cafus Curvarum ubi ipfius z
index 0 augetur vel diminuitur perpetua additione vel
fubdudione quantitatis ». Calus alter eft Curva*
rum ubi index ^ augetur vel diminuitur unitatibus.

Bbb C^S.

[i86]

CAS. II.

Ordinatae pz^'^R'^ & qz^l'^-'R'^, quibus arese pA
& qB jam refpondeant, (i in R.feu e-|-fz'!'|-gz^" du-
cantur ac deinde ad R viciffim applicentur, eva-
dunt pe -)- pfzi \' pgz^" x z^'^R^-' & qez" -(- qfz^o
-j^qgz3" X z^-'R'^"'. Et per Prop. III. eft azSR'^
area Curvae cujus Ordinata eft fl^ie iJ^afz''r[;J^^agz^''
in z^-'R'^"* , & bz^'l "R'^ area Curvae cujus ordinata
eft ^j.Jbez" Iljbfz^" '/_Jbgz3'' in z^-'R^'^ Et harum qua-

tuor arearum lumma eft pA-|-qB-(-az^R''-l-bz^l''R'^
oc lumma refpondentium ordinatarum

+pe

Hafz" li^aaz^"

TM -1-2 AH D

+pf

+ qe

+ qf

I bgz3« in z^-^R'

2AH

+qg

A-I

Si terminus primus tertius & quartus ponantur fe-
orfim aequales nihilo, per primum iiet 9ae-l-pe = o
feu — fla = p, per quartum — 6b — nb - i^wb = q , & per

tertium (eliminando p & q) "t = b. Unde fecundus
fit '"^ "^^^"^g^ adeoq;fumma quatuor Ordinatarum eft
"^^^'z^+^-'R^'S&fumma totidem refpondentium
arearurn eft ^^^^'''\-^-fz^^'r>YiK-^2ik—:^^^■^^^g'^•

VI-

C 187 ] ^^^

Dividantur hx fumms per '^ — r^^ & fi Quotum
pofterius dicatur D, erit D area curvae cujus ordi-
nata eil Quotum prius z^^'^'^R^'' . Et eadem ratione
ponendo omnes Ordinatae terminos praeter primum^
aequales nihilo poteft area Curvas inveniri cujus Or-
dinata eft z^'^R'^"'. Dicatur area ifta C, & qua ra-
tione ex areis A & B inventas funt areae C ac D, ex
his areis C ac D inveniri poflTunt alia duae E & F
ordinatis z^'^R^'^ & z^'l'"''R'^"^ congruentes, & fie de-
inceps in infinitum. Et per analyfin contrariam
regredi licet ab areis E & F ad areas C ac D, &
inde ad areas A & B, aliafi:j; quae in progreflione fe-
quuntur. Igitur fi index ^ perpetua unitatum ad-
ditione vel fubdudione augeatur vel minuatur, Sc
ex areis quae Ordinatis fie prodeuntibus refpondent
duae fimpliciffimae habentur ; dantur aliae omnes in
infinitum. Q. E. O.

C ^ S. III.

Et per cafus holce duos conjun6tos, fi tam in^
dex a perpetua additione vel lubdudtione ipfius",
quam index x perpetua additione vel fubdu^tione
unitatis, utcunq; augeatur vel minuatur, dabuntur
areae fingulis prodeuntibus Ordinatis refpondentes.
Q. E. O.

Bbbi Cu4S.

[i88]

C A S. IV.

Et fiinili augmento fi ordinata conftat ex qua-
tuor nominibus in vinculo radicaii ^^ dantur tres
arearum, vel fi conftat ex quinq; nominibus &
dantur quatuor arearum, & fie deinceps : dabun-
tur areoe omnes qua? addendo vel fubducendo nume-
rum n indici 5 vel unitatem indici x generari pofTunt.
Et par eft ratio Curvarum ubi ordinatae ex binomiis
conflantur, & area una earum quae non funt geome-
trice quadrabiles datur. Q. E. O.

PROP. VIII. THEOR. VI.

Si pro e4-fz»-l-gz^«'|-&c. & k -[- lz»-l-mz^-l-&c.
fcribantur R & S ut fupra,& in Curvae alicujus Or-
dinata z^-'^^R'^l'^S'^'l'' maneant quantitates datae 9,
», A^ M, e, f, g, k, 1, m, &c. & pro <^, t^ &: k, fcri-
bantur lucceffive numeri quicunq; integri : & fi
dentur areae duarum ex curvis quae per ordinatas
fie prodeuntes defignantur fi quantitates R & S funt
binomia, vel fi dentur areae trium ex curvis fi R
&. S conjun<5lim ex quinq; nominibus conftant, vei
areae quatuor ex curvis fi R & S conjundim ex fex
nominibus conftant, 8c fie deinceps in infinitum :
dico quod dabuntur areae curvarum omnium.

Demonftratur ad modum Propofitionis fuperioris.

PROP.

[iSpJ

PROP. IX. THEOR. VII.

JEquantur Curvarum areoe inter fe quarum Or-
dinatae i'unt reciproce ut fluxioncs AbfcifTarura.

Nam contenta Tub Ordinatis & fluxion ibus Ab-
fciflanim erunt aequalia, & fluxiones arearuin ilint
ut hafc contenta.

CO ROL. T.

Si affumatur relatio qusevis inter Abfciflas dua-
rum Curvarum, & inde per Prop. i. quaeratur
relatio fluxionum Abfciflarum, & ponantur Ordi-
natae reciproce proportionates fluxionibus, inveniri
pofTunt innumers Curvae quarum areas fibi mutuo
aequales erunt.

CO ROL. II.

Sic enim Curva omnis cujus haec eft Ordinata
z^' in e -[- fz»-^gz^" -|- &c.|'' aflumendo quantitatem
quamvis pro » & ponendo ^1=9 & z^ = x, migrat in
aliam fibi aequalem cujus ordinata eft tj^''ir:« in
e-j-fx''4-gx2''^-.&c7|'^.

CO^

[ ^9<=> ]

COR.OL. III.

Et Curva omnis cujiis Ordinata eft z^*' in

a ■-]- bz« -|- cz^" -J- &c. X e-j-fzw-l-gz^" &c.|^,a{rumen-
do quantitatem quamvis pro " & ponendo a^s &
Z^ = X, migrat in aliam libi cequalem cujus ordinata
eft -'x'-^ in a 4- bx'-i- cx^^ t &c. xe -j- tx'-)- gx^' -[- &c.|^-

COROL. IV.

Et Curva omnis cujus Ordinata eft z^'' in
a~l- bz""-l- cz'« -^ &c. X e -|- fz" -|- gz^" -\- &c. j''
X k -|- Iz" -[- mz^" -1" 6cc.p^ afTumendo quantitatem
quamvis pro v & ponendo J = s & z' = x, migrat in
aliam fibi aequalem cujus ordinata eft !" xfc in a-l-bxv
■-y-cx"''h^^- ^e + tx''-(-gx^''-l-&c.J^xk-j-Ix''4-mx^''4-&c.f

COROL. V.

Et Curva omnis cujus Ordinata eft z^' in

e'^f z» -\- gz^" -\- ^cJ'^ ponendo i= x migrat in

aliam fibi aequalem cujus ordinata eft -^^ ^ e-|-f3C»

xp^^qr^h id eft x5TTT;^^^M=^f ^^^"^^""^

nomina in vinculo radicis vel ^6-^-i-|-„^ x g-\'^^'*'\' ^^^1

fi tria funt nomina ; & fie deinceps.

CO-

[Ipl ]

COROL. VI.

Et Curva omnis cujus Ordinata eft z^' In
e ■-]- f z" -'(- gz^" -(- &c.('' X k '\-lz"-\- inz'"-|-&c.|'^
poiiendo z = x migrat in aliam fibi cequalem cu-
jus ordinata eft —f^~, x e -|- fx" -|- gx''" -|- .:^c.j^

xk-i-ix-".-|-mx--''-|-&c.|'^ id eft ^p^^zf;;^^ x fl^l'
xl-|-kx''|'^ fi bina funt nomlna in vinculis radicum,
vel x^T^'i 2«H-«/. X g -j- fx'-j- ex'^p x l-\-kx"j^ fi tria

funt nomina in vinculo radicis prions ac duo in
vinculo pofterioris : & (ic in aliis. Et nota quod
areae duge aequales in noviffimis hifce duobus Co-
roUariis jacent ad contrarias partes ordinatarum.
Si area in alterutra curva adjacet abiciiToe , area
huic cequalis in altera curva adjacet abfcifTse pro-
duda*.

COROL. VII.

Si relatio inter Curvge alicujus Ordinatam y &
Abfciflam z definiatur per aequationem quamvis
fedtam hujus forma?,y« in e -|- fy«z^-|- gy^^z^^-l- hy^^'z^'^
+ Sec. = z^ in k -j- ly"z^ -]- my^^z^^ -\- Sec ha-c
figura alTumendo s^-^^, x = -^z^ & x=^^}^, migrat
ifl aliam fibi sequalem cujus Abfcifla x, ex data

Ordinata

[192]

Ordinata v, determinatur per aquationcm non
affeftam ,-v*'^ x e-|- fv"i- gv"" -^ hv^" -\- Scc.i'^ x k -)- Iv

COROL. VIII.

Si relatio inter Curvse alicujus Ordinatam y &
Ablciiram z definitur per gquationem quamvis
aifedam hujiis formce, y* in e-|-fy"Z'^']-gy^''z^'^ ]-8lc.
= z^ in k-l-ly»z'^.43'm^2=r:p§^,_|_z7inp4^qyV

Jp ry^^z^^-j-Scc. haec figura aflumendo s = '^,x= '-z',

^=7=7 & " = -;;=:j-, migrat in aliam fibi asqualem
cujus Abfcifla x ex data Ordinata v determinatur
per aequationem minus affedam v* in e-|- fv-j-gv^*

-\- &c. = si'x'^ in k '\- Iv" -[- mv^" -1- &c. -\- sV in
p-j-qv"-)- rv"^" -|- &c.

COROL. IX.

Curva omnis cujus Ordinata eft ttz*' in

e:|;i^'z":|4gz"'i" &c. X e^-fz"-!- gz2«&c.h X
\d - - b lez" -i fz-i » -I- gz"'!""'-! &c.lf, li fit 9 = ^x &
affumantur x - ez' ^^ fz"+H -^- gzH-2« J^- 8cc.| " , -^^i
& '-^ = Jiz:r\ migrat in aliam fibi squalem cujus ordi^
iiata eft x-^ xa -j-bx'^l ^ Et nota quod ordinata prior

[193].

in hoc Corollario evadit {implicior ponendo x'= i,
vel ponendo ^ = i Sc efficiendo ut radix dignitatis
extrahi poffit cujus index eft «, vel etiam ponendo
«= — I & ^ = I = T = ^ =7r , ut alios cafus prate-
ream.

COR.OL. X.

Pro ez" '\- fzH-» J^ gz'+^w _|^ Slc, ^ez"-' j:||fz'^'*-'
+2« gz"'''^"'' -1- &<^- k + Iz" + niz^" '1- &c. & "Iz"-'
+ 2.3mz^»-'-l-&c. fcribantur R, r, S oc s refpe6tive, &
Curva omnis cujus ordinata eft ^rSr -j-r ? Rs in R'^'' S«*'*
x~aSM^^bRl , ft fit«=!i'' = "- = ?, 1 = 0-, ^-* = *,

& R'S?* = X, migrat in aliam fibi aequalem cujus or-
dinata eft X* X a-^bx^l". Et nota quod Ordinata
prior evadit fimplicior, ponendo unitates pro t, y,
& ^ vel fc, & faciendo ut radix dignitatis extrahi
poffit cujus index eft «, vel ponendo « = —i vel

PROP. X, PROB. III.

Invenire figuras fimpliciffimas cum quibus Curva
qucevis geometrice compari poteft, cujus ordinatim
applicata y per oequationem non affedam ex data ab-
icifta z determinatur.

c c CAS,

I ml

C A S. I.

Sit Ordinata az^', & area erit a az^, ut ex Prop.V.
ponendob = o = c = d = f— g = h &e=i, facile col-
ligitur.

CAS. II.

Sit Ordinata az^-' x e-)-fz«-l- gz^f'' -]- kc. & fi
curva cum figuris rediiineis geometrice coinparari
poteft, quadrabitur per Prop. V. ponendo h = o-c
— d. Sin minus convertetur in aliam curvam fibi
aequalem cujus Ordinata eft -^x^^ x e-l-fx-|-gx'&c.|''''
per Corol. a. Prop. IX. Deinde fi de dignitatum
mdicibus 9ji & ^_i per Prop. VII. rejiciantur uni-
tates donee dignitates ilia? fiant quam minims, de-
venietur ad figuras fimpliciffimas quce hac ratione
coUigi poflunt. Dein harum unaquaeq; per Corol. 5.
Prop. IX. dat aliam quse nonnunquam limplicior
eft. Et ex his per Prop. III. & Corol. 9 & 10,
Prop. IX. inter le collatis, flgura:adhuc fimpliciores
quandoq; prodeunt. Deniq; ex figuris fimplicif-
fimis affumptis fado regreffu computabitur area
qucefita.

CAS.

C A S. III.

Sit Ordinata z«-' x a -\^ bz" 4- cz^" -\-. &.c.

X e -(- fz" -1" gz'" -]- ^cl'^-' , & haec figura fi quadrari
poteft, quadrabitur per Prop. V. Sin minus, di-
ftinguenda eft ordinata in partes z^-^ x a x e -j- f z*
+ gz^" -1- ^c.|^-', z^-' X bz" X e-J- fz«-|-gz^«-|-&c.H,
&c. & per Caf. 2. inveniendae funt figurae limpli-
ciffimce cum quibus figurae partibus illis refpon-
dentes comparari poflunt. Nam areae figura rum
partibus illis refpondentium Tub fignis fuis -|- & — •
conjundse component aream totam quaefitam.

CAS. IV.

Sit Ordinata z^'' x a -j- b z" + c z^" -\~ &c. x
e_|-.fz« -'pgz'" -]- &C.H X k -1- Iz^-^mz'-'i-^cclt*-':
& fi Curva quadrari poreft,quadrabitur per Prop. VI.
Sin minus, convertetur in fimpliciorem per Corol.4.-
Prop. IX. ac deinde comparabitur cum figuris fim-
pliciffimis per Prop. VIII. 8c Corol. 6, ^ 8c 10.
Prop. IX. ut fit in Cafu a & 3.

C A S. V.

^ *

Si Ordinata ex variis partibus conftat , partes
fingulse pro ordinatis curvarum totidem habendae
runt,& curvae illae quotquot quadrari pofTunt^figilla-

C c c 2 tim

tim quadrandae funt, earumq; ordlnatiTS de ordlrtata
tota demendje. Dein Curva quam ordinatce pars
relidua defignat fcorfim ( lit in Calli 2, ^ & ^^"^
cum tiguris limpliciffimis comparanda eft cum qui-
bus comparari poteft. Et fumma arearum omnium
pro area Curva^ propofitic habenda eft.

COROL. I.

Hinc etiam Curva .omnis cujus Ordinata eft ra-
dix quadratica atfeda cequationis luce, cum figuris
limpliciffimis feu redilineis leu curvilineis com'
pari poteft. Nam radix ilia ex duabus partibus
lemper conftat quas feorfim fpeftatae non funt aequa-
num radices affedae. Proponatur cequatio aayy
"l- zzyy = ^a'y -^-^z^y— z"*, & extra(5ta radix erit
__ a' -\-' z5+ a\/a''-^'az'— z'' cujus pars rationalis

J aa -\- zz .

a:?4-z? o • • 1- aVa" + 2az^ - z'
aa^zz & pars irrationalis jjipjs lunt

ordinate curvarum quse per banc Propofitionem

vel quadrari pofTunt vel cum figuris limpliciffimis

comparari cum quLbus collationem geometricam ad-

raittunt.

[COROL. 11.

Et curva omnis cujus Ordinata per aequationem
quamvis affedam definitur quae per Corol. 7. Prop.
IX. in cequationem non alfed:am migrat, vel qua-

dratur

dratur per hanc Propofitionem fi quadrari poteft vel
comparatur cum figurls fimpliciffimis cum quibus
compari poteft. Ethac rationeCurva omnis quadni-
tur cujus aequatio eft trium terminorum. Nam squa-
tio ilia ii affeda fie tranimutatur in non affedam per
Corol.y. Prop.IX. ac delude per Corol. a & 5. Prop.
IX. in fimplicftimam migrando, dat vel quadratu-
ram figurae fi quadrari poteft, vel curvam fimplicil- -
fimam quacum comparatur.

COROL. 111.

Et Curva omnis cujus Ordinata per cequacionem
quamvis affedam definitur quce per Corol. 8. Prop.
IX. in sequationem quadraticam atfedam migrat;
vel quadratur per banc Propofitionem & hujus Co-
rol. I . fi quadrari poteft, vel comparatur cum figu-
ris fimplieiflimis cum quibus collationem geometri-
camadmittit.

SCHOLIUM.

Ubi quadranda? funt figurae; ad Regulas hafce
generales Temper recurrere nimis moleftum effet :
praeftat Figuras quae fimpliciores funt & magis ufui
efTe poflunt femel quadrare & quadraturas in Ta-
bulam referre, deinde Tabulam confulere quoties
ejufmodi Curvam aliquam quadrare oportet. Hu-
jus autem generis funt Tabulae duae fequentes, in
quibus z denotat AbfcifTam, y Ordinatam redan-

gulam

gulam & t AreamCurvae quadrandae, Scd, e, f, g^
g. h, " funt quantitates datge cum fignis fuis-J^ & — .

TABULA

Curvarumjimpliciorum qua qmdrari pojfunt.

Cur varum formae. Cur varum arese.

Forma prima.

dz«-' = y. »z" = t

Forma fecunda.
dz*^^ dzn — d _^

Forma tertia,

I . dz." Ve-j-fz^^y. f„f R' = t, exiftente R = V^^^fe"

a. dz!? Ve-j-fz^^y. ""^^«ff '" dR' =t.
3.dz?;'Ve~l^=y. -^— gg-3o^^-^dR3^t.

Forma quarta.

dz«-i ad

^•-7=F = y- ^R=t.

d7^«"' I -

dz^"'

[199]

A 73M-1 j6ee— 8efz„-|- 6ttz2„

^•-=::=y' 777. ^R=^f-

^24ii-i — 9'5e3-l-4Seefz«— 36effz2)rl-3of32:„

TABULA

Curvarunt fimpliciorum cjua cum Kllipjl &'
Hyper Ma compart poffunt.

Sit jam aGD vel PGD vel GDS Sedlo
Conica cujus area ad Quadraturam Curvas pro- f^i' ^j'^s??^.
pofitge requiritur, fitq; ejus centrum A, Axis Ka,
Vertex a, Semiaxis conjugatus AP, datum Abfciflae
principium A vel a vel «, AbicitHi A B vel a B vel
aB = x, Ordinata re^languia BD = v, & Area
A B DP vel aBDG vel aBDG = s, exiftente «G Or-
dinata ad punftum «. Jungantur KD, AD, aD. Du-
catur Tangens DT occurrens Abfcififoe AB in T,
& compleatur parallelogrammum ABDO. Et
fiquando ad quadraturam Curvs propofitas requi-
runtur ares duarum Sedionem Conicarum, dica-
tur pofterioris AbfcilTa I, Ordinata t^ & Area <r.
Sit autem -^ differentia duarum quantitatum ubi in-
certum eft utrum pofterior de priori an prior de po-
fteriori iiibduci debeat.

Curva-

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In Tabulis hifce, feries Cur varum cujufq; format.
utrinq; in infinitum continuari poted. Scilicet
in Tabula prima, in numeratoribus arearurn for-
ma? tertise & quarts, numeri coefficientes initialium
terminorum (2, — 4,16, — 96, 868,&c.) generan-
tur multiplicando numeros — 2, — 4., — 6, — io,&c.
in fe continuo, & fubfequentium. terminorum coef-
ficientes ex initialibus derivantur multiplicando
ipibs gradatim, in Forma quidem tertia , per — '-
—l^—h—h —To &<^"- ill quarta vero per— i,— i''
— h — h — L> &c. Et Denominatorum coefficientes
5, 15, 105, &c. prodeunt multiplicando numeros
I, ^, 5, 7, 9, &c. in fe continuo.

In fecunda vero Tabula, feries Curvarum forma
primae, fecund^, quintt^e, fextse, nonse & decims ope
folius divifionis, & formse reliquce ope Propofitio-
nis tertia: & quarts, utrinq; producuntur in in*
finitum.

Quinetiam hx feries- mutando iignum numeri „
variari folent. Sic enim, e. g. Curva IVc-j-fz"— v
evadit ~- \/i-\-tz».

PROP. IX. THE OR. VIIL

'Sit A DIG Curva quae vis AbfcifTam habensir^^ p-
AB=z ScOrdinatam BD=y, 8c fit AEKC Curva
alia cujus Ordinata BE squalis eft prioris ares

ABC

[20(5]

ADB ad unitatem applicatas , &AFLC Curva
tertia cujus Ordinata B F aequalis eft lecundae areoe
AEB ad unitatem applicatae, & AGMC Curva
quarta cujus Ordinata B G aequalis eft tertia? ares
AFB ad unitatem applicats, &AHNC Curva
quinta cujus Ordinata BH aequalis eft quarta? areae
AGB ad unitatem applicatoe, & lie deinceps in
intinitum. Et lunto A, B, C, D, E, &c. Arei^ Cur-
varum Ordinatas habentium. y, zy, z'y, z^y, z''y,
& AbfcifTam communem z.

Detur Abfciffa quasvis AC=t, fitq; BC=t — z
= x, & lunto P, Q, R, S, T arese Curvarum Ordi-
natas habentiumx, xy, xxy, x'y, x''y & AbfcilHim
communem x.

Terminenter autem hx areae omnes ad Abfciflam
totam datam A C, nee non ad Ordinatam poiitione
datam & infinite produdam C I : & erit arearum
fub initio politarum prima A D I C=A=P, lecunda
AEKC=:t A— B=Q.Tertia AFLC = ^^^=f2±^ - fR.
Quarta AGMC = Ii±=in|b££zLD==is. Quinta

A H N C UA— 4f;B-|-6t(:C— 4tD -4- E i -t-

CO-

[ 207 ]

COROL.

llnde fi Curv32 quarum Ordlnata* liint y, zy,
z'y, z^y, &c. vel y, xy, x'y, x'y, &c. quadrari
polTunt, quadrabuntur etiam Curvas ADIC, AEKC,
AFLC, AGMC,&c. & habebuntur OrdinatiE BE,
BF, BG, BH areis Curvarum propoitionales.

SCHOLIUM.

QuaiititatLim fluentium fluxiones efife prlmas ^
fecundas, tertias, quartas , aliafq; diximus fupra.
Hce tluxiones lunt ut termini ferierum infinita-
rum con vei gentium. Ut fi z"fit quantitas fluens &
fluendo evadat z-i-o|", deinde refolvatur in feriem
convergentem z"i »oz«-''|- ~-ool »-^4~ "^-~^''"^-t-^'!Q^z''-3

■■\-Slc. terminus primus hujus feriei z" erit quan-
titas ilia tluens, lecundus moz"'' erit ejus incremen-
tum primum feu ditferentia prima cui nafcenti pro-
portionalis eft ejus fluxio prima , tertius ^ oz"*^
erit ejus incrementum fecundum feu ditferentia fe-
cunda cui nalcenti proportionalis eft ejus fluxio
fecunda, quartus "^"^^'"^ ^"■o^z'^^ erit ejus incremen-
tum tertium feu diiferentia tertia cui nafcenti
fluxio tertia proportionalis eft, & fie deinceps in
infinitum.

[ Exponi

[ 208 ]

Exponi autem poflunt hcefiuxloncs perCurvarum
Ordinatas BD, BE, BF, BG, BH, Sec. Ut ii
Ordinata BE (=^) fit quantitas fluens, erit
ejus fluxio prima ut ordinata B D. Si B F (=M?)
fit quantitas iiuens, erit ejus fluxio prima ut Or-
dinata BE Sc fluxio fecunda ut Ordinata BD. Si
BH (=~) fit quantitas fluens, erunt ejus fluxio-
lies, prima, fecunda, tertia & quarta, ut Ordinate
BG, BF, BE, BDrefpeaive.

Et hinc in aequationibus quse quantitates tantum
duas incognitas involvunt, quarum una efl: quan-
titas uniformiter fluens Sc altera efl: fluxio quaelibet
quantitatis alterius fluentis , inveniri poteft fluens
ilia altera per quadraturam Curvarum. Exponatur
enim fluxio ejus per Ordinatam B D, & fl hxc lit
fluxio prima, quaTatur area ADB=BExi, (i
■fluxio iecunda , quaeratur area AEB = BFx i, li
fluxio tertia, quaeratur area AFB = BGxi,&c.
Be area inventa erit exponens fluentis quaeflt^.

Sed Sc in sequationibus quae fluentem & ejus
fiuxionem primam fine altera fluente , vel duas
if'jufdem fluentis fluxiones, primam & fecundam,
vel fecundam & tertiam, vel tertiam & quartam,
Sec. flne alterutra fluente involvunt : inveniri pof-
funt fluentes per quadraturam Cui*varum. Sit

2equatio aav = av ~\-y vv , exiflente v = B E ,

v = B D, z =A B & z = I , & sequatio ilia com-

plendo dimenfiones fluxionum, evadet aav = avz

4VVZ5 feu jTqrr^ =z. Jam fluat v uniformiter &

fit

[ 209 ]

fit ejus fluxio v=i & erit -^=i^ & quadrando

Curvain cujus Ordinata eft ^7^:7^ & Abfciffa v, ha-

• • • •

bebitur liuens z. Adha?c fit aequatio aav=av-|-vv
exiftente v=BF, v=BE, v=BD & z=AB Sc
per relationem inter v&vfeuBD &BE invenie-
tur relatio Inter A B & B E ut in exemplo fuperiore.
Deinde per banc relationem invenietur relatio in-
ter A B & BF quadrando Curvam AEB.

^quatlones quse tres incognitas quantitates invol-
vunt aliquando reduci pofTunt ad crquationes quK
duas tantum involvunt, & in his cafibus fluentes
invenientur ex fluxionibus ut fupra. Sit sequatio
a — bx'":=cxy«y -j-dy^'iyy. Ponatur y«y=v & erit
a — bx'\xv-|-dvv. Ha^c aequatio quadrando Cur-
vam cujus Abiciffa eft x & Ordinata v dat aream
V, & oequatio altera y''y=v regrediendo ad fluentes
dat ^y*'"^"^ =v. Unde habeturfluens y.

Quinetiam in cequationibus quae tres incognitas
involvunt & ad aequationes quae duas tantum in-
volvunt reduci non poflunt, fluentes quandoq;
prodeunt per quadraturam Curvarum. Sit aequatio
a x"^^-- bxxp' = r ex'""' y' .-|- s ex'^y y^'^ — fy y', exiftente

X = I . Et pars pofterior r e x'""^ y ' -j' s e x^ y y ^'' — f y y ',
regrediendo ad fluentes, fit exry' — J_.-yti-i^ quoe

proinde eft ut area Curvs: cujus AbicilHi eft x &
Ordinata ax^"-J hxf^ & inde datur fluens y.

E e e Sit

[210]

Sit aequatio x x a x"i -1- hxf =— zzi* Et fluens

cujus fluxio eft X X ax'^'i-bx''^ erit 'ut area Curvs
cujus AblcilTa eft x & Ordinata eft a'x'" -[- bx„f.
Item fluens cuius fluxio eft iHiii g^t ut area Curvae

cuius AbfcilTaeft y & Ordinata 'IZ!^, id eft
(per Cafum i- Formse (],uart3s Tab. I.) ut area
^7f Ve-j-fy- Pone ergo '^^- Ve-j-^fy sequalem areae

Cuvvx cujus Ablcifla eft x & Ordinata ax"'^-^ bx«'|^
Sc habebitur fluens y,

Et nota quod, fluens omnis quoe ex fluxione prima
GoUigitur augeri poteft vel minui quantitate quavis
non fluente. Qua: ex fluxione fecunda colligitur
augeri poteft vel minui quantitate quavis cujus
fluxio iecunda nulla eft. Quae ex fluxione tertia
colligitur augeri poteft vel minui quantitate quavis
cujus fluxio tertia nulla eft. Et fie deinceps in in-
finitum^.

Poftquam vero fluentes ex fluxionibus colleds
funt, fi de veritate Conclufionis dubitatur, fluxio-
nes fluentium inventarum viciflim colligendse funt
& cum fluxionibus fub initio propofitis comparanda?.
Nam fi prodeunt cequales Conclufio rede fe ha-

bet :

[2II]

bet : fin minus , corrigendse funt fluentes fie , lit
earuni fluxiones fluxionibus lub initio propofitis
aequentur. Nam tSc Fluens pro lubitu aflumi po-
teft & affumptio corrigi ponendo fluxionem flii-
entis aflumptae iequalem fluxioni propofita?, & tcr-
minos homologos inter le comparando.

Et his principiis via ad majora fternitur.

F I N I S.

£ RR JT A
BOOK I. OfOptkkj.

PArt I. p.3. 1.20. rropertiei vahhh, ib.p.";. I.5. mi tlut C, p.6. I.9. Z>E, p.2J. I.23.
are two %n, p.27. 1.5. ;» t/v M.?rg(n }'"t fitj.i4 C? 1 5' p.30-l-7- ^■'V, '.?. M, p.
44. 1.15. *f rv^propofed, p.52. 1. 17. if ;'4;'<;'' OVc/i,', p.57. l.ulr. emerging, p.6c. I.25.
contain rvith the, p.64. I.18. 4Hi I4f''. p.65. I.13. <rf fk, p.66. 1.3.J>Vwn7rn(/^r, p.67.
1.25.Ce»fer, 1,31. 4I Inches, p.6S. l.B. ro 16, 1.9. o>- 5^, ^.-jiA.i.bifea, p.72.1.13.
/rffo, 1.20. if/Kg. PartII.p.86.1.5./tf/o;)/>e.<^, p.89. 1.9.»wi/e i)', p.93. l.iS. w 771^
1.28,29, *^ f/^f" f*'>.i ^xio/n 0/ tlv firfl Tart of this Book, the Lam, p.105. l.'^.fee rcpre-
fented, p. 144. 1. 24, i, ,% h, f, ro, r^,?- P- ^S. i^P- for i/Z,.i. i/i.2. write
rarti.rm7. p.i22. 1.9. iH^/Vo, P- 130' l-ip-*" ^he Angle, p.132. 1.6. bj the bright-
jnefs, p.135* 1.14. for i/wffcf, l.ie.^r/^ P^rt yoH, p.136. \.26. prfl Part, I.27. lights,
p.137. 1.20. gww, accordingly m, p. 138. 1. 21. rrop.6. Pjrt.2. p. 139. 1.5. ow which,
l\i4.2A.i7. xr which hdve been, p.i43' ^•7-pwple, \.\6. feveral Lights,l.2^. of white.

BOOK. II.

P.5. 1. '^Micely tJbf,p. 7. 1.9.>', t d^fJOte, I.28. rfew ;i/wrr,p. 10. 1. 24. igoo to 1024,
\\iiA.ii.oli^uities,I. p.i?- 1.4- Hj W9> p.25.1. 11. i o±, p.31.1. 12. more com-
,poiwded, p.';5. l.^.^xes reflet, I.24. ami therefore their Colours arife, ^.6<,.\.e,. corpuf-
cks can, p.71. 1.17. given breadth, p.84. 1. 4- ^''^ tothofe, p. 96. 1. 24. Ohfervation of
thif Tart of this Book, p.103. 1.17. tr^r fot^e thicknefs, p. 105. 1. 19. o/t/;u 7v/vfc; J{h!g,
p. 107. 1.20. become e^ual to the third oj thafe.

Eniimerntio Linearum.

■^.\MA.20.daiisfignisfuis, p. 144- 1.27. re/J-ia/wt, p. 146. \.<^. fmt Afpnpxoto, p.
I '54. l.i 3. cx-\-i dar Ordinatamy - , l.H- qux generatur.

Oitadrattira Cur'varum.

■ p.i68.1.24.>-efli^B,p.i76.1.ult. ^ l^'iZ»,^.i%i.\.ii.it,b,c,i^c. e,f,g,^c. 1,1, m,

e<f._p. 185. 1.4. /« z9-i, p.i88. 1.14- zfl + MO-, p. 190. 1. i^.vel » ^.jr~^^
p. 192. 1.18. g'ji'~+-2»). P.I93.1.U. aSui-bRr, *'•

7 \.\v

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K A T ^^"7 B X T A ^i" B N A

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Book H. Pi

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( inrnnmi.

iirvai'um Tdlr. IJ.

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( un>(irni}} Iiih /]'

C (ifonniDi Til/'. H.

Cunmnim TalrJ.

Ciirvani/n Tab. M.

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Full text of "Opticks: or, A treatise of the reflections, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures"

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